L3 Integration
Question Bank
The diagram shows part of the curve $y = 2 + \dfrac{6}{x^2}$.
Find the area of the shaded region, between the curve, the $x$-axis and the lines $x = 1$ and $x = 3$.
You must use calculus and show the results of any integration needed to solve the problem.
The diagram shows the curve $y = 6\cos 3x$ for $0 \le x \le \dfrac{\pi}{6}$.
Find the area of the shaded region between the curve and the $x$-axis.
You must use calculus and show the results of any integration needed to solve the problem.
The diagram shows the curve $y = e^{2x} + 3$.
Find the area of the shaded region, between the curve, the $x$-axis, the $y$-axis and the line $x = 1$.
Give your answer in exact form.
You must use calculus and show the results of any integration needed to solve the problem.
The diagram shows part of the curve $y = \sqrt{2x + 5}$.
Find the area of the shaded region, between the curve, the $x$-axis and the lines $x = 2$ and $x = 10$.
You must use calculus and show the results of any integration needed to solve the problem.
The diagram shows the curve $y = x^2 - 4x + 5$ and the line $y = x + 1$.
Find the area of the shaded region enclosed between them.
You must use calculus and show the results of any integration needed to solve the problem.
The diagram shows part of the curve $y = \dfrac{6}{\sqrt{x}}$. It meets the line $y = 6$ at $A$ and the line $y = 2$ at $B$.
The shaded region lies between the curve and the $x$-axis, from the vertical line through $A$ to the vertical line through $B$. Find its area.
You must use calculus and show the results of any integration needed to solve the problem.
The diagram shows part of the curve $y = 4e^{-0.5x}$.
The shaded region, between the curve, the $x$-axis, the $y$-axis and the line $x = k$, has an area of 6 square units.
(a) Find the exact value of $k$.
(b) Explain why, however large $k$ is, the area of this region is always less than 8 square units.
You must use calculus and show the results of any integration needed to solve the problem.
Find the total area of the regions between the curve $y = x^3 - 4x$ and the $x$-axis, from $x = 0$ to $x = 3$.
You must use calculus and show the results of any integration needed to solve the problem.
The diagram shows the curves $y = x^2 - 2x$ and $y = 6x - x^2$.
Find the area of the shaded region enclosed between them.
You must use calculus and show the results of any integration needed to solve the problem.
The diagram shows the curve $y = \dfrac{4}{2x + 1}$.
The region between the curve, the $x$-axis, the $y$-axis and the line $x = 4$ is split by the line $x = k$ into two parts, $A$ and $B$, of equal area.
Find the value of $k$.
You must use calculus and show the results of any integration needed to solve the problem.
The diagram shows the curve $y = \sqrt{4x + 1}$ and the line $y = x + 1$.
Find the area of the shaded region enclosed between them.
You must use calculus and show the results of any integration needed to solve the problem.
The diagram shows the curve $y = 4\sin 2x$ for $0 \le x \le \dfrac{\pi}{2}$.
The shaded region, between the curve, the $x$-axis and the line $x = k$, has an area of 3 square units.
Find the exact value of $k$.
You must use calculus and show the results of any integration needed to solve the problem.
The diagram shows the curve $y = e^{2x}$ and the line $y = 8$, which meet at $P$.
Find the area of the shaded region, bounded by the curve, the line $y = 8$ and the $y$-axis.
Give your answer in exact form.
You must use calculus and show the results of any integration needed to solve the problem.
The curve $y = \dfrac{6}{x + 1}$ and the line $y = 6 - x$ meet at two points, as shown.
Find the area of the shaded region enclosed between them. Give your answer in exact form.
You must use calculus and show the results of any integration needed to solve the problem.
The region between the curve $y = 3x^2$, the $x$-axis and the lines $x = k$ and $x = k + 2$ has an area of 26 square units.
Find the possible values of $k$.
You must use calculus and show the results of any integration needed to solve the problem.
Find the total area of the regions between the curve $y = e^x - 2$ and the $x$-axis, from $x = 0$ to $x = 2$.
Give your answer in exact form.
You must use calculus and show the results of any integration needed to solve the problem.
The curve $y = x^3 - 2x^2$ and the line $y = 3x$ enclose two regions, shaded in the diagram.
Find the total area of the two shaded regions.
You must use calculus and show the results of any integration needed to solve the problem.
The diagram shows the curve $y = 4\cos^2 x$ and the line $y = 1$, for $-\dfrac{\pi}{2} \le x \le \dfrac{\pi}{2}$.
Find the exact area of the shaded region enclosed between them.
You must use calculus and show the results of any integration needed to solve the problem.
The region enclosed by the parabola $y = x^2$ and the line $y = 4$ is split into two parts of equal area by the line $y = k$, where $0 < k < 4$.
Find the exact value of $k$.
You must use calculus and show the results of any integration needed to solve the problem.
The diagram shows the curves $y = e^{2x}$ and $y = e^x$.
The shaded region, between the two curves, the $y$-axis and the line $x = k$ (where $k > 0$), has an area of 8 square units.
Find the exact value of $k$.
You must use calculus and show the results of any integration needed to solve the problem.
The diagram shows the curves $y = x^n$ and $y = x^{1/n}$ for $x \ge 0$, where $n > 1$.
(a) Show that the area of the region enclosed between the curves is $\dfrac{n - 1}{n + 1}$.
(b) Find the value of $n$ for which the enclosed region has an area of $\dfrac{3}{4}$.
You must use calculus and show the results of any integration needed to solve the problem.
The tangent to the curve $y = 2e^{0.5x}$ at the point $P$ passes through the origin, as shown.
(a) Find the coordinates of $P$.
(b) Find the exact area of the shaded region, bounded by the curve, the tangent and the $y$-axis.
You must use calculus and show the results of any integration needed to solve the problem.
The curves $y = 2e^x$ and $y = 7 - 3e^{-x}$ meet at two points, as shown.
Find the exact area of the shaded region enclosed between them.
You must use calculus and show the results of any integration needed to solve the problem.
The tangent to the curve $y = x^2$ at the point $(-1, 1)$ is $y = -2x - 1$, and the tangent at $(3, 9)$ is $y = 6x - 9$. The two tangents meet at the point $T(1, -3)$, as shown.
Find the area of the shaded region, which is bounded by the curve and the two tangents.
You must use calculus and show the results of any integration needed to solve the problem.
The diagram shows the curves $y = \cos x$ and $y = \cos 2x$ for $0 \le x \le \pi$.
Find the exact area of the shaded region enclosed between the curves.
You must use calculus and show the results of any integration needed to solve the problem.
The diagram shows the curve $y = 2\sqrt{x}$, the line $y = 12 - 2x$ and the curve $y = \dfrac{x^2}{3} - 2x$.
Find the area of the shaded region, which is bounded by all three.
You must use calculus and show the results of any integration needed to solve the problem.
The line $y = mx$, where $0 < m < 4$, divides the region enclosed by the curve $y = 4x - x^2$ and the $x$-axis into two parts of equal area, as shown.
Find the exact value of $m$.
You must use calculus and show the results of any integration needed to solve the problem.
The diagram shows the curves $y = \sin 2x$ and $y = \tan x$ for $0 \le x < \dfrac{\pi}{2}$.
Find the exact area of the shaded region enclosed between them.
You must use calculus and show the results of any integration needed to solve the problem.
The diagram shows the line $y = 2x$, the curve $y = \dfrac{2}{x}$ for $x > 0$, and the curve $y = x^2 - 3$.
Find the exact area of the shaded region, which is bounded by all three.
You must use calculus and show the results of any integration needed to solve the problem.
The line $y = kx$, where $0 < k < 2$, meets the curve $y = \dfrac{2x}{x^2 + 1}$ at the origin and at a point $P$ with $x > 0$, as shown.
Show that the area of the shaded region enclosed between the curve and the line is
$ \ln\frac{2}{k} + \frac{k}{2} - 1 $
You must use calculus and show the results of any integration needed to solve the problem.