A region in the –plane is enclosed by the –axis and the curve for
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
A region in the xy–plane is enclosed by the y–axis and the curve x=4−y2 for −2≤y≤2. A student claims that the area of this region can be found by evaluating a single definite integral with respect to y.
(a) Describe, without performing any calculations, how a horizontal slice of this region can be used to construct a definite integral that represents the area.
(b) Explain why integrating with respect to y requires only one integral in this context, whereas integrating with respect to x would require splitting the region into separate subregions.
(c) Interpret the meaning of the integrand in the context of the geometry of the region and justify the limits of integration.