Question

set up a definite integral that represents the indicated shaded area over the interval $c,d$.
choose the correct answer below.
$\bigcirc$ a. $\int_{c}^{d}p(x)-k(x) d x$
$\bigcirc$ b. $\int_{c}^{d}k(x)-p(x) d x$
$\bigcirc$ c. $\int_{c}^{d}-p(x)+k(x) d x$
$\bigcirc$ d. $\int_{c}^{d} p(x) d x$
$\bigcirc$ e. $\int_{c}^{d} k(x) d x$
$\bigcirc$ f. $\int_{c}^{d}p(x)+k(x) d x$

Explanation:

Step1: Recall the formula for the area between two curves

The area \( A \) between two curves \( y = f(x) \) and \( y = g(x) \) over the interval \([a,b]\) is given by \( A=\int_{a}^{b}|f(x)-g(x)|dx \). When \( f(x)\geq g(x) \) for all \( x\in[a,b] \), then \( A = \int_{a}^{b}[f(x)-g(x)]dx \).

Step2: Identify the upper - and lower - bound functions

In the interval \([c,d]\), from the graph, we can see that \( y = k(x) \) is the upper - bound function and \( y = p(x) \) is the lower - bound function.

Answer:

B. \( \int_{c}^{d}[k(x)-p(x)]dx \)