• A Level Maths

Demo

• A Level
• Numerical Integration

1. 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. $\dfrac{77}{60}$

2. 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. $\dfrac{15}{2}$

3. 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

$0.635$

4. 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

5. 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.

$4.85$

6. 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.

$2.51$

7.

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

8.

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. $31.2$

9.

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. $0.312$

10.

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

11.

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. $14.9$

12.

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. $2(16\ln2 + 4.47) = 31.1$