1.08b Integrate x^n: where n != -1 and sums

Find \(\int 7x^3 \, dx\). [3]

Oskar is designing a building. Fig. 12 shows his design for the end wall and the curve of the roof. The units for \(x\) and \(y\) are metres. \includegraphics{figure_12}

1. Use the trapezium rule with 5 strips to estimate the area of the end wall of the building. [4]
2. Oskar now uses the equation \(y = -0.001x^3 - 0.025x^2 + 0.6x + 9\), for \(0 \leq x \leq 15\), to model the curve of the roof.
   1. Calculate the difference between the height of the roof when \(x = 12\) given by this model and the data shown in Fig. 12. [2]
   2. Use integration to find the area of the end wall given by this model. [4]

Fig. 9 shows the cross-section of a straight, horizontal tunnel. The \(x\)-axis from 0 to 6 represents the floor of the tunnel. \includegraphics{figure_9} With axes as shown, and units in metres, the roof of the tunnel passes through the points shown in the table.

\(x\)	0	1	2	3	4	5	6
\(y\)	0	4.0	4.9	5.0	4.9	4.0	0

The length of the tunnel is 50 m.

1. Use the trapezium rule with 6 strips to estimate the area of cross-section of the tunnel. Hence estimate the volume of earth removed in digging the tunnel. [4]
2. An engineer models the height of the roof of the tunnel using the curve \(y = \frac{x}{81}(108x - 54x^2 + 12x^3 - x^4)\). This curve is symmetrical about \(x = 3\).
   1. Show that, according to this model, a vehicle of rectangular cross-section which is 3.6 m wide and 4.4 m high would not be able to pass through the tunnel. [2]
   2. Use integration to calculate the area of the cross-section given by this model. Hence obtain another estimate of the volume of earth removed in digging the tunnel. [5]

\includegraphics{figure_3} Figure 3 shows part of the curve \(y = \text{f}(x)\) where $$\text{f}(x) = \frac{1 - 8x^3}{x^2}, \quad x \neq 0.$$

1. Solve the equation \(\text{f}(x) = 0\). [3]
2. Find \(\int \text{f}(x) \, dx\). [3]
3. Find the area of the shaded region bounded by the curve \(y = \text{f}(x)\), the \(x\)-axis and the line \(x = 2\). [3]

Given that $$\frac{dy}{dx} = 3\sqrt{x} - x^2,$$ and that \(y = \frac{4}{3}\) when \(x = 1\), find the value of \(y\) when \(x = 4\). [7]

The gradient of a curve is given by \(\frac{dy}{dx} = \frac{6}{x^3}\). The curve passes through \((1, 4)\). Find the equation of the curve. [5]

A curve has gradient given by \(\frac{dy}{dx} = 6x^2 + 8x\). The curve passes through the point \((1, 5)\). Find the equation of the curve. [4]

The gradient of a curve is given by \(\frac{dy}{dx} = 4x + 3\). The curve passes through the point \((2, 9)\).

1. Find the equation of the tangent to the curve at the point \((2, 9)\). [3]
2. Find the equation of the curve and the coordinates of its points of intersection with the \(x\)-axis. Find also the coordinates of the minimum point of this curve. [7]
3. Find the equation of the curve after it has been stretched parallel to the \(x\)-axis with scale factor \(\frac{1}{2}\). Write down the coordinates of the minimum point of the transformed curve. [3]

Find \(\int 7x^2 dx\). [3]

The gradient of a curve is given by \(\frac{dy}{dx} = \frac{18}{x} + 2\). The curve passes through the point \((3, 6)\). Find the equation of the curve. [5]

The gradient of a curve is given by \(\frac{dy}{dx} = 6x^{\frac{1}{2}} - 5\). Given also that the curve passes through the point \((4, 20)\), find the equation of the curve. [5]

Find \(\int_2^5 (2x^3 + 3) dx\). [3]

The gradient of a curve is given by \(\frac{dy}{dx} = 6\sqrt{x} - 2\). Given also that the curve passes through the point \((9, 4)\), find the equation of the curve. [5]

Find \(\int_2^5 \left(1 - \frac{6}{x^3}\right) dx\). [4]

Find \(\int_1^2 (12x^5 + 5) dx\). [4]

The gradient of a curve is \(3\sqrt{x} - 5\). The curve passes through the point \((4, 6)\). Find the equation of the curve. [5]

A curve has gradient given by \(\frac{dy}{dx} = 6\sqrt{x}\). Find the equation of the curve, given that it passes through the point \((9, 105)\). [4]

Find \(\int_1^2 \left(x^4 - \frac{3}{x^2} + 1\right) dx\), showing your working. [5]

Find \(\int 30x^2 dx\). [3]

Find \(\int (x^5 + 10x^3) dx\). [4]

Find \(\int (3x^5 + 2x^{-\frac{1}{2}}) dx\). [4]

Fig. 11 shows the curve \(y = x^3 - 3x^2 - x + 3\). \includegraphics{figure_11}

1. Use calculus to find \(\int_{-1}^{3} (x^3 - 3x^2 - x + 3) dx\) and state what this represents. [6]
2. Find the \(x\)-coordinates of the turning points of the curve \(y = x^3 - 3x^2 - x + 3\), giving your answers in surd form. Hence state the set of values of \(x\) for which \(y = x^3 - 3x^2 - x + 3\) is a decreasing function. [5]