1.08e Area between curve and x-axis: using definite integrals

10 \includegraphics[max width=\textwidth, alt={}, center]{5201a3d5-7733-4d10-9de5-0c2255e3ad60-18_401_584_264_776} The diagram shows part of the curve \(y = \frac { 1 } { 2 } \left( x ^ { 4 } - 1 \right)\), defined for \(x \geqslant 0\).

1. Find, showing all necessary working, the area of the shaded region.
2. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
3. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(y\)-axis.

10 \includegraphics[max width=\textwidth, alt={}, center]{518bb805-5b14-4b41-94fd-38a31a90c218-18_551_689_260_726} The diagram shows part of the curve \(y = \sqrt { } ( 5 x - 1 )\) and the normal to the curve at the point \(P ( 2,3 )\). This normal meets the \(x\)-axis at \(Q\).

1. Find the equation of the normal at \(P\).
2. Find, showing all necessary working, the area of the shaded region.

8 \includegraphics[max width=\textwidth, alt={}, center]{17ca6dd2-271b-4b06-8433-354493feaf06-12_485_570_262_790} The diagram shows parts of the graphs of \(y = 3 - 2 x\) and \(y = 4 - 3 \sqrt { } x\) intersecting at points \(A\) and \(B\).

1. Find by calculation the \(x\)-coordinates of \(A\) and \(B\).
2. Find, showing all necessary working, the area of the shaded region.

7 \includegraphics[max width=\textwidth, alt={}, center]{d178603a-f59a-4986-b5ab-b47eceedb2fc-10_503_853_260_641} The diagram shows part of the curve with equation \(y = k \left( x ^ { 3 } - 7 x ^ { 2 } + 12 x \right)\) for some constant \(k\). The curve intersects the line \(y = x\) at the origin \(O\) and at the point \(A ( 2,2 )\).

1. Find the value of \(k\).
2. Verify that the curve meets the line \(y = x\) again when \(x = 5\).
3. Find, showing all necessary working, the area of the shaded region.

10 \includegraphics[max width=\textwidth, alt={}, center]{567c3d72-c633-4ae0-8605-f63f93d718c4-18_979_679_262_731} The diagram shows part of the curve \(y = 1 - \frac { 4 } { ( 2 x + 1 ) ^ { 2 } }\). The curve intersects the \(x\)-axis at \(A\). The normal to the curve at \(A\) intersects the \(y\)-axis at \(B\).

1. Obtain expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\int y \mathrm {~d} x\).
2. Find the coordinates of \(B\).
3. Find, showing all necessary working, the area of the shaded region.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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1. Use the trapezium rule with three intervals to find an approximation to \(\int _ { 1 } ^ { 4 } \frac { 6 } { 1 + \sqrt { x } } \mathrm {~d} x\). Give your answer correct to 5 significant figures.
2. Find the exact value of \(\int _ { 1 } ^ { 4 } 2 \mathrm { e } ^ { \frac { 1 } { 2 } x - 2 } \mathrm {~d} x\).
3. \includegraphics[max width=\textwidth, alt={}, center]{2d6fc4c5-70ec-4cd8-9b48-59d5ce0e39b7-11_556_805_262_705} The diagram shows the curves \(y = \frac { 6 } { 1 + \sqrt { x } }\) and \(y = 2 \mathrm { e } ^ { \frac { 1 } { 2 } x - 2 }\) which meet at a point with \(x\)-coordinate 4. The shaded region is bounded by the two curves and the line \(x = 1\). Use your answers to parts (a) and (b) to find an approximation to the area of the shaded region. Give your answer correct to 3 significant figures.
4. State, with a reason, whether your answer to part (c) is an over-estimate or under-estimate of the exact area of the shaded region.

3 \includegraphics[max width=\textwidth, alt={}, center]{ed12a4fb-e3bf-4d00-ad09-9ba5be941dd5-04_531_739_258_703} The diagram shows the curve with equation \(y = 3 \sin x - 3 \sin 2 x\) for \(0 \leqslant x \leqslant \pi\). The curve meets the \(x\)-axis at the origin and at the points with \(x\)-coordinates \(a\) and \(\pi\).

1. Find the exact value of \(a\).
2. Find the area of the shaded region.

3 \includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-04_776_483_310_769} The diagram shows the curve with equation \(y = 8 \mathrm { e } ^ { - x } - \mathrm { e } ^ { 2 x }\). The curve crosses the \(y\)-axis at the point \(A\) and the \(x\)-axis at the point \(B\). The shaded region is bounded by the curve and the two axes.

1. Find the gradient of the curve at \(A\). \includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-04_2715_35_141_2011}
2. Show that the \(x\)-coordinate of \(B\) is \(\ln 2\) and hence find the area of the shaded region.

3 \includegraphics[max width=\textwidth, alt={}, center]{8beee722-7f86-454a-bc36-27e83f1483fd-04_684_455_260_845} The diagram shows the curve \(y = 2 + \mathrm { e } ^ { - 2 x }\). The curve crosses the \(y\)-axis at the point \(A\), and the point \(B\) on the curve has \(x\)-coordinate 1 . The shaded region is bounded by the curve and the line segment \(A B\). Find the exact area of the shaded region.

3 \includegraphics[max width=\textwidth, alt={}, center]{b4a4082c-f3cd-47c5-8673-680dae9a22bd-04_684_455_260_845} The diagram shows the curve \(y = 2 + \mathrm { e } ^ { - 2 x }\). The curve crosses the \(y\)-axis at the point \(A\), and the point \(B\) on the curve has \(x\)-coordinate 1 . The shaded region is bounded by the curve and the line segment \(A B\). Find the exact area of the shaded region.

6 \includegraphics[max width=\textwidth, alt={}, center]{68f4b2dc-a05d-4061-aaf0-de15cfe186a9-08_616_531_269_799} The diagram shows the curves \(y = \frac { 6 } { 3 x + 2 }\) and \(y = 3 \mathrm { e } ^ { - x } - 3\) for values of \(x\) between 0 and 4. The shaded region is bounded by the two curves and the lines \(x = 0\) and \(x = 4\). Find the exact area of the shaded region, giving your answer in the form \(\ln a + b + c \mathrm { e } ^ { d }\).

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1. Find the exact area of the region bounded by the curve \(y = 1 + \mathrm { e } ^ { 2 x - 1 }\), the \(x\)-axis and the lines \(x = \frac { 1 } { 2 }\) and \(x = 2\).
2. \includegraphics[max width=\textwidth, alt={}, center]{e3ee4932-8219-4332-9cd2-e7f835522469-3_469_719_397_753} The diagram shows the curve \(y = \frac { \mathrm { e } ^ { 2 x } } { \sin 2 x }\) for \(0 < x < \frac { 1 } { 2 } \pi\), and its minimum point \(M\). Find the exact \(x\)-coordinate of \(M\).

4 \includegraphics[max width=\textwidth, alt={}, center]{3b217eb4-3bd3-4800-a913-749754bf109f-2_524_625_1425_758} The diagram shows the curve \(y = \mathrm { e } ^ { x } + 4 \mathrm { e } ^ { - 2 x }\) and its minimum point \(M\).

1. Show that the \(x\)-coordinate of \(M\) is \(\ln 2\).
2. The region shaded in the diagram is enclosed by the curve and the lines \(x = 0 , x = \ln 2\) and \(y = 0\). Use integration to show that the area of the shaded region is \(\frac { 5 } { 2 }\).

6 \includegraphics[max width=\textwidth, alt={}, center]{3b217eb4-3bd3-4800-a913-749754bf109f-3_401_586_817_778} The diagram shows part of the curve with equation $$y = 4 \sin ^ { 2 } x + 8 \sin x + 3$$ and its point of intersection \(P\) with the \(x\)-axis.

1. Find the exact \(x\)-coordinate of \(P\).
2. Show that the equation of the curve can be written $$y = 5 + 8 \sin x - 2 \cos 2 x$$ and use integration to find the exact area of the shaded region enclosed by the curve and the axes.

9 \includegraphics[max width=\textwidth, alt={}, center]{208eab3e-a78c-43b4-918f-a9efc9b4f47a-4_429_748_264_699} The diagram shows part of the curve \(y = \frac { x } { x ^ { 2 } + 1 }\) and its maximum point \(M\). The shaded region \(R\) is bounded by the curve and by the lines \(y = 0\) and \(x = p\).

1. Calculate the \(x\)-coordinate of \(M\).
2. Find the area of \(R\) in terms of \(p\).
3. Hence calculate the value of \(p\) for which the area of \(R\) is 1 , giving your answer correct to 3 significant figures.

8 \includegraphics[max width=\textwidth, alt={}, center]{5b219e1c-e5a0-4f75-910d-fca9761e5088-3_435_895_799_625} The diagram shows the curve \(y = 5 \sin ^ { 3 } x \cos ^ { 2 } x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\).

1. Find the \(x\)-coordinate of \(M\).
2. Using the substitution \(u = \cos x\), find by integration the area of the shaded region bounded by the curve and the \(x\)-axis.

5 \includegraphics[max width=\textwidth, alt={}, center]{4c71f68a-efb9-4408-bf03-874e0d4426d5-2_458_807_1786_667} The diagram shows the curve $$y = 8 \sin \frac { 1 } { 2 } x - \tan \frac { 1 } { 2 } x$$ for \(0 \leqslant x < \pi\). The \(x\)-coordinate of the maximum point is \(\alpha\) and the shaded region is enclosed by the curve and the lines \(x = \alpha\) and \(y = 0\).

1. Show that \(\alpha = \frac { 2 } { 3 } \pi\).
2. Find the exact value of the area of the shaded region.

10 \includegraphics[max width=\textwidth, alt={}, center]{772393d7-6e81-4b99-913a-63c9f87d1af2-16_524_689_260_726} The diagram shows the curve \(y = \sin 3 x \cos x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\) and its minimum point \(M\). The shaded region \(R\) is bounded by the curve and the \(x\)-axis.

1. By expanding \(\sin ( 3 x + x )\) and \(\sin ( 3 x - x )\) show that $$\sin 3 x \cos x = \frac { 1 } { 2 } ( \sin 4 x + \sin 2 x ) .$$
2. Using the result of part (i) and showing all necessary working, find the exact area of the region \(R\).
3. Using the result of part (i), express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\cos 2 x\) and hence find the \(x\)-coordinate of \(M\), giving your answer correct to 2 decimal places.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

10 \includegraphics[max width=\textwidth, alt={}, center]{e26f21c5-3776-4c86-8440-6959c5e37486-18_337_529_260_808} The diagram shows the curve \(y = ( \ln x ) ^ { 2 }\). The \(x\)-coordinate of the point \(P\) is equal to e, and the normal to the curve at \(P\) meets the \(x\)-axis at \(Q\).

1. Find the \(x\)-coordinate of \(Q\).
2. Show that \(\int \ln x \mathrm {~d} x = x \ln x - x + c\), where \(c\) is a constant.
3. Using integration by parts, or otherwise, find the exact value of the area of the shaded region between the curve, the \(x\)-axis and the normal \(P Q\).

6 \includegraphics[max width=\textwidth, alt={}, center]{79efa364-da5a-4888-85a9-dc4de1e0908e-3_543_825_287_660} The diagram shows the curve \(y = ( 3 - x ) \mathrm { e } ^ { - 2 x }\) and its minimum point \(M\). The curve intersects the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).

1. Calculate the \(x\)-coordinate of \(M\).
2. Find the area of the region bounded by \(O A , O B\) and the curve, giving your answer in terms of e.

6 \includegraphics[max width=\textwidth, alt={}, center]{dd7b2aee-4318-48e8-97c0-541e47f2e83a-2_551_567_1416_788} In the diagram, \(A\) is a point on the circumference of a circle with centre \(O\) and radius \(r\). A circular arc with centre \(A\) meets the circumference at \(B\) and \(C\). The angle \(O A B\) is \(\theta\) radians. The shaded region is bounded by the circumference of the circle and the arc with centre \(A\) joining \(B\) and \(C\). The area of the shaded region is equal to half the area of the circle.

1. Show that \(\cos 2 \theta = \frac { 2 \sin 2 \theta - \pi } { 4 \theta }\).
2. Use the iterative formula $$\theta _ { n + 1 } = \frac { 1 } { 2 } \cos ^ { - 1 } \left( \frac { 2 \sin 2 \theta _ { n } - \pi } { 4 \theta _ { n } } \right)$$ with initial value \(\theta _ { 1 } = 1\), to determine \(\theta\) correct to 2 decimal places, showing the result of each iteration to 4 decimal places. \(7 \quad\) Let \(\mathrm { f } ( x ) = \frac { 2 x ^ { 2 } - 7 x - 1 } { ( x - 2 ) \left( x ^ { 2 } + 3 \right) }\).
3. Express \(\mathrm { f } ( x )\) in partial fractions.
4. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).

10 \includegraphics[max width=\textwidth, alt={}, center]{5d76d7cb-c2cb-4fa4-8133-d5d43702b293-3_366_764_1914_687} The diagram shows the curve \(y = \frac { x ^ { 2 } } { 1 + x ^ { 3 } }\) for \(x \geqslant 0\), and its maximum point \(M\). The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = p\).

1. Find the exact value of the \(x\)-coordinate of \(M\).
2. Calculate the value of \(p\) for which the area of \(R\) is equal to 1 . Give your answer correct to 3 significant figures.

10 \includegraphics[max width=\textwidth, alt={}, center]{efa44efb-9350-4d54-b5e1-4f722781a5f3-3_366_764_1914_687} The diagram shows the curve \(y = \frac { x ^ { 2 } } { 1 + x ^ { 3 } }\) for \(x \geqslant 0\), and its maximum point \(M\). The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = p\).

1. Find the exact value of the \(x\)-coordinate of \(M\).
2. Calculate the value of \(p\) for which the area of \(R\) is equal to 1 . Give your answer correct to 3 significant figures.

7 \includegraphics[max width=\textwidth, alt={}, center]{67a12825-d7ce-4853-ada4-b8d3009331b5-3_531_759_262_694} The diagram shows the curve \(y = \mathrm { e } ^ { - x }\). The shaded region \(R\) is bounded by the curve and the lines \(y = 1\) and \(x = p\), where \(p\) is a constant.

1. Find the area of \(R\) in terms of \(p\).
2. Show that if the area of \(R\) is equal to 1 then $$p = 2 - \mathrm { e } ^ { - p }$$
3. Use the iterative formula $$p _ { n + 1 } = 2 - \mathrm { e } ^ { - p _ { n } }$$ with initial value \(p _ { 1 } = 2\), to calculate the value of \(p\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

5 \includegraphics[max width=\textwidth, alt={}, center]{9e1bd528-e7c4-4936-a05a-dde1d1ace7c2-2_512_775_1318_683} The diagram shows the curve \(y = \cos x\), for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). A rectangle \(O A B C\) is drawn, where \(B\) is the point on the curve with \(x\)-coordinate \(\theta\), and \(A\) and \(C\) are on the axes, as shown. The shaded region \(R\) is bounded by the curve and by the lines \(x = \theta\) and \(y = 0\).

1. Find the area of \(R\) in terms of \(\theta\).
2. The area of the rectangle \(O A B C\) is equal to the area of \(R\). Show that $$\theta = \frac { 1 - \sin \theta } { \cos \theta }$$
3. Use the iterative formula \(\theta _ { n + 1 } = \frac { 1 - \sin \theta _ { n } } { \cos \theta _ { n } }\), with initial value \(\theta _ { 1 } = 0.5\), to determine the value of \(\theta\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.