CAIE IGCSE Additional Math 14.13 Evaluate Definite Integrals and Areas Questions

Practise using definite integrals to find exact areas and related geometric quantities bounded by curves, straight lines and coordinate axes in Cambridge Additional Mathematics problems.

Exam points

Use definite integrals to calculate exact areas and related geometric quantities bounded by curves, lines and axes.

[Maximum number: 10]

Figure for Question CAIE IGCSE Additional Math 14.13 Evaluate Definite Integrals and Areas Questions question 1 — CAIE IGCSE Additional Math

The diagram shows parts of the graphs of $y=2+5e^x$ and $y=4-3e^{2x}$ .
Find the area of the shaded region.
Give your answer in the form $a+b\ln3$ , where a and b are exact constants.

Writes as a 3-term quadratic in $e^x$ : $3e^{2x}+5e^x-2=0$ soi
Solves/factorises their quadratic in $e^x$ $e^x=\frac13$ [ $e^x=-2$ ]
$x=\ln\frac13=-\ln3$ only solution, nfww
Correct plan: $\int_{\ln(1/3)}^0(3e^{2x}+5e^x-2)\,dx$
Integral $\frac32e^{2x}+5e^x-2x$
Correct substitution of limits
$\frac{14}{3}-2\ln3$
M2 dep on all previous marks;
M1 for one correct final term.

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CAIE IGCSE Additional Math 14.13 Evaluate Definite Integrals and Areas Questions question 1