MATH200 April 2012
• Q1 (a) • Q1 (b) • Q2 (a) • Q2 (b) • Q3 • Q4 (a) • Q4 (b) • Q4 (c) • Q5 (a) • Q5 (b) • Q6 (a) • Q6 (b) • Q7 • Q8 • Q9 • Q10 •
Question 06 (b)

Evaluate $I$ from part (a) if
 $f(x,y)={\frac {\exp(x)}{2x}}$

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1

Using the solution of part (a), we calculate the integral
$\displaystyle \int _{1}^{2}\int _{x^{2}}^{x+2}{\frac {\exp(x)}{2x}}\;{\text{d}}y{\text{d}}x$
It make sense to use the order of integration, where we first integrate with respect to $\displaystyle y$, since integration with respect to $\displaystyle x$ first is very hard with this integrand.

Hint 2

After integration with respect to $\displaystyle y$, you need to integrate
 $\displaystyle \int _{1}^{2}{\frac {\exp(x)}{2x}}\left(x+2x^{2}\right){\text{d}}x$
Factorize the term in the brackets, then use integration by parts.

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Solution

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Using our answer from part (a), we can write I with the given ƒ as
 $I=\int _{1}^{2}\int _{x^{2}}^{x+2}{\frac {\exp(x)}{2x}}\,dydx.$
We can evaluate the integral as follows:
 ${\begin{aligned}I&=\int _{1}^{2}\int _{x^{2}}^{x+2}{\frac {\exp(x)}{2x}}\,dydx\\&=\int _{1}^{2}{\frac {\exp(x)}{2x}}\left(x+2x^{2}\right)\,dx\\&=\int _{1}^{2}{\frac {\exp(x)}{2x}}(x+2)(x+1)\,dx\\&=\int _{1}^{2}\exp(x)(x+1)\,dx.\end{aligned}}$
Using integration by parts, with u=x+1 and dv = exp(x),
 ${\begin{aligned}I&=\int _{1}^{2}\exp(x)(x+1)\,dx\\&=\exp(x)(x+1){\Big \vert }_{1}^{2}\int _{1}^{2}\exp(x)\,dx\\&=3\exp(2)\left(\exp(x){\Big \vert }_{1}^{2}\right)=2\exp(2)+\exp(1)\end{aligned}}$
Therefore,
 ${\color {blue}\int _{1}^{2}\int _{x^{2}}^{x+2}{\frac {\exp(x)}{2x}}\,dydx=2e^{2}+e^{1}}$

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MER QGH flag, MER QGQ flag, MER QGS flag, MER QGT flag, MER Tag Multiple integral, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag

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