Integration

The diagram shows the curve with equation $y = 5x^{\frac{3}{2}} - 20x$ and the line with equation $y = x - 16$. The $x$-coordinates of the points of intersection of the curve and line are 1 and 16.

Find the area of the shaded region between the curve and the line.

The diagram shows the curve with equation $y = \frac{9}{(5x+4)^{\frac{1}{2}}}$ and the line $y = 6 - 3x$. The line and the curve intersect at the point $P$ which has $y$-coordinate 3.

Find the area of the shaded region.

Given that $\int_1^3 \left( \frac{a}{(4x-3)^2} + 2 \right) dx = 12$, find the value of the constant $a$.

The equation of a curve is such that $\frac{\mathrm{d}y}{\mathrm{d}x} = 12(2x - 5)^2 + 8x$. It is given that the curve passes through the point $(2, 4)$.

Find an equation of the curve.

The diagram shows part of the curve $y = x^2 - \frac{1}{x^2}$. The shaded region is bounded by the curve, the line $x = 2$ and the $x$-axis.

Find the volume formed when the shaded region is rotated through $360^\circ$ about the $x$-axis, giving your answer correct to 2 decimal places.

The diagram shows part of the curve with equation $y = \frac{1}{2}x + \frac{4}{x}$ and the line $y = 4.5$.

Find the exact volume of the solid formed when the shaded region is rotated through $360^\circ$ about the $x$-axis.

Find the exact value of

A curve is such that $\frac{dy}{dx} = 3(4x + 5)^{\frac{1}{2}}$. It is given that the points $(1, 9)$ and $(5, a)$ lie on the curve.

Find the value of $a$.

The diagram shows part of the curve with equation $y = \frac{1}{(5x - 4)^{\frac{1}{3}}}$ and the lines $x = 2.4$ and $y = 1$. The curve intersects the line $y = 1$ at the point $(1, 1)$.

Find the exact volume of the solid generated when the shaded region is rotated through $360^\circ$ about the $x$-axis.