Definite Integrals & Area — practice questions

Evaluate $\displaystyle\int_1^4 \left(2x + \sqrt{x}\right)dx$, giving an exact answer.

Evaluate $\displaystyle\int_0^{\pi/3}\left(\sin 2x + \cos x\right)dx$, giving an exact answer.

Find all values of $k$ for which $\displaystyle\int_0^k \left(x^2 - 4x\right)dx = 0$, and explain what the non-zero solution means geometrically.

Find the area between the curve $y = 4 - x^2$ and the $x$-axis, between the two points where the curve crosses the axis.

Find the total area enclosed between the curve $y = x^2 - 3x$ and the $x$-axis for $0 \leq x \leq 5$.

The region bounded by the curve $y = \sqrt{x}$, the $x$-axis and the line $x = a$ (where $a > 0$) has area 18 square units. Find $a$, and show that the line $x = a$ divides this region into two parts of equal area at $x = a/\sqrt[3]{4}$.

Find the area enclosed between the curve $y = 6x - x^2$ and the line $y = x$.

Find the area of the region enclosed between $y = e^x$, $y = e^{-x}$ and the line $x = 2$.

The line $y = mx$ (with $m > 0$) divides the region enclosed between the curve $y = 4x - x^2$ and the $x$-axis into two parts of equal area. Find the exact value of $m$.

Use the trapezium rule with three strips to estimate $\displaystyle\int_1^4 \frac{1}{x}\,dx$, giving your answer to 3 decimal places.

Use the trapezium rule with four strips to estimate $\displaystyle\int_0^{\pi}\sin x\,dx$, and compare your estimate with the exact value, explaining the difference.

The speed of a vehicle is recorded every 10 seconds: at $t = 0, 10, 20, 30, 40, 50, 60$ seconds the speeds are $0, 8, 15, 19, 21, 22, 22$ m s⁻¹. Estimate the distance travelled in the first minute, state the units, and discuss whether your estimate is likely to be too high or too low.