Q1
Answer
The lower bound of the area under the graph of $y=\dfrac{1}{x}$ between $x=1$ and $x=5$ can be approximated using rectangles of width $1$.
  1. Write down this approximation using Sigma notation.
  2. Evaluate it.
  1. $\displaystyle\sum_{n=2}^5 \dfrac{1}{n}$
  2. $\dfrac{77}{60}$
Q2
Answer
For $$\int_0^{2.5} 2x^2 \mathrm{d}x$$ Use rectangles of width $0.5$ to find:
  1. its upper bound;
  2. its lower bound.
  1. $\dfrac{55}{4}$
  2. $\dfrac{15}{2}$
Q3
Answer
Use the trapezium rule with 4 equally spaced strips to estimate the area of the region bound by the curve $y=\dfrac{1}{1+\sqrt{x}}$, the coordinate axes and the line $x=1$, giving your answer to 3 decimal places
$0.635$
Q4
Answer
Given $\mathrm{f}(1) = 9$, $\mathrm{f}(2.25) = 17$, $\mathrm{f}(3.5) = 25$, $\mathrm{f}(4.75) = 21$, and $\mathrm{f}(6) = 13$,
  1. Use the trapezium rule with all the values given to find an estimate for the integral $\displaystyle\int_1^6\mathrm{f}(x)\ \mathrm{d}x$
  2. Given that $\mathrm{f}(x)$ is a smooth curve, state whether your answer is an over or under estimate of the actual area.
  1. $92.5$
  2. Concave, underestimate
Q5
Answer
Use the trapezium rule with 4 equally spaced strips to estimate the value of $$\displaystyle\int_0^4 \dfrac{2^x}{x+2}\ \mathrm{d}x$$ correct to 3 significant figures.
$4.85$
Q6
Answer
Use the trapezium rule with 5 equally spaced strips to estimate the value of $$\displaystyle\int_1^3 (\sqrt{x}-\log x)^2\ \mathrm{d}x$$ correct to 3 significant figures.
$2.51$
Q7
Answer
  1. Estimate $\displaystyle\int_0^{\frac{\pi}{3}} \mathrm{e}^{\tan^2x}\ \mathrm{d}x$ using the trapezium rule with 4 equally spaced strips.
  2. Hence estimate $\displaystyle\int_0^{\frac{\pi}{3}} \mathrm{e}^{\sec^2x}\ \mathrm{d}x$
  3. Are the estimates are likely to be over or underestimates?
  4. Are the estimates are likely to be accurate?
  1. $4.12$
  2. $11.2$
  3. Convex, overestimate
  4. Large errors possible because of large jump between $y$ values of last two strips
Q8
Answer
  1. Estimate $\displaystyle\int_0^2 2^{\sqrt{x}}\ \mathrm{d}x$ using the trapezium rule with 4 equally spaced strips.
  2. Hence estimate $\displaystyle\int_0^2 2^{\sqrt{x}} + 3\ \mathrm{d}x$
  3. Hence estimate $\displaystyle\int_0^2 2^{\sqrt{x}+3}\ \mathrm{d}x$
  1. $3.90$
  2. $9.90$
  3. $31.2$
Q9
Answer
  1. Estimate $\displaystyle\int_0^{\frac{\pi}{3}} \cos^2 x\ \mathrm{d}x$ using the trapezium rule with 4 equally spaced strips.
  2. Hence estimate $\displaystyle\int_0^{\frac{\pi}{3}} \sin^2 x\ \mathrm{d}x$
  1. $0.735$
  2. $0.312$
Q10
Answer
  1. Estimate $\displaystyle\int_0^{\frac{\pi}{3}} \sqrt{\tan x}\ \mathrm{d}x$ using the trapezium rule with 4 equally spaced strips.
  2. Will the answer increase or decrease if more strips are used?
  1. $0.769$
  2. Mostly concave so increases with more strips
Q11
Answer
  1. Estimate $\displaystyle\int_0^1 \mathrm{e}^{-x^2}\ \mathrm{d}x$ using the trapezium rule with 4 equally spaced strips.
  2. Hence estimate $\displaystyle\int_0^1 \mathrm{e}^{-x^2+3}\ \mathrm{d}x$
  1. $0.743$
  2. $14.9$
Q12
Answer
  1. Estimate $\displaystyle\int_2^{18} \ln\dfrac{2}{\sqrt{4+\sqrt{x}}}\ \mathrm{d}x$ using the trapezium rule with 4 equally spaced strips.
  2. Hence estimate $\displaystyle\int_2^{18} \ln(4+\sqrt{x})\ \mathrm{d}x$
  1. $-4.47$
  2. $2(16\ln2 + 4.47) = 31.1$