Question

6 which of the following represents the area of the shaded region in the figure above?
a) \\(\int_{a}^{b} f(x) dx\\)
b) \\(\int_{a}^{b} f(x) - d dx\\)
c) \\(\int_{a}^{b} f(x) - a dy\\)
d) \\(\int_{c}^{d} a - f(y) dy\\)
e) \\(\int_{c}^{d} f(y) - a dy\\)

Explanation:

To determine the area of the shaded region, we analyze the figure. The shaded region is bounded vertically (along the \( y \)-axis) from \( y = a \) to \( y = d \), and horizontally, for each \( y \), the right boundary is \( x = f(y) \) and the left boundary is \( x = a \). The area between two curves (or a curve and a line) with respect to \( y \) is given by the integral of (right - left) with respect to \( y \). So, the width at each \( y \) is \( f(y)-a \), and we integrate this from \( y = a \) to \( y = d \). Thus, the area is \( \int_{a}^{d}[f(y)-a]dy \), which corresponds to option e.

Answer:

e) \( \int_{a}^{d}[f(y)-a]dy \)