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Contents Index Calc Dark Mode Prev Up Next \(\newcommand{\R}{\mathbb{R}}
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Activity 12.3.7 .
Consider the iterated integral
\(\int_{x=0}^{x=4} \int_{y=x/2}^{y=2} e^{y^2} \, dy \, dx\text{.}\)
(a) Explain why we cannot find a simple antiderivative for
\(e^{y^2}\) with respect to
\(y\text{,}\) and thus are unable to evaluate
\(\int_{x=0}^{x=4} \int_{y=x/2}^{y=2} e^{y^2} \, dy \, dx\) in the indicated order using the Fundamental Theorem of Calculus.
(b) Given that
\(\iint_D e^{y^2} \, dA = \int_{x=0}^{x=4} \int_{y=x/2}^{y=2} e^{y^2} \, dy \, dx\text{,}\) sketch the region of integration,
\(D\text{.}\)
(c) Rewrite the given iterated integral in the opposite order, using
\(dA = dx \, dy\text{.}\) (Hint: You may need more than one integral.)
(d) Use the Fundamental Theorem of Calculus to evaluate the iterated integral you developed in (c). Write one sentence to explain the meaning of the value you found.
(e) What is the important lesson this activity offers regarding the order in which we set up an iterated integral?
