AP Calculus BC 8.5: Area with Definite Integrals

Practise AP Calculus BC 8.5 by setting up definite integrals for planar regions, switching between horizontal and vertical slices, and checking equivalent area expressions.

Exam points

Identify equivalent area integrals by comparing vertical and horizontal slices of the same region.

[Maximum number: 1]

Let R be the region bounded by the graph of $y=\ln x$ , the vertical line $x=e^{2}$ , and the x-axis, as shown in the figure. Which of the following integrals are equal to the area of R ?

I. $\int_{1}^{e^{2}} \ln x d x$
II. $\int_{0}^{2} e^{y} d y$
III. $\int_{0}^{2}\left(e^{2}-e^{y}\right) d y$

I only

III only

I and II only

I and III only

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CHA-5.A—Calculate areas in the plane using the definite integral—Topic 8.5 question 1