Question

homework 7 additional integration topics
hw. 7.1 area under the curve

1. refer to fig. 7 - 1. set up definite integrals that represent the indicated shaded areas over the given intervals.

(a) over interval a, b
(b) over interval 0, c
(c) over interval c, d
(d) over interval a, d

Explanation:

Step1: Recall area - under - curve formula

The area \(A\) under the curve \(y = f(x)\) over the interval \([m,n]\) is given by \(A=\int_{m}^{n}|f(x)|dx\). If \(f(x)\geq0\) on \([m,n]\), then \(A = \int_{m}^{n}f(x)dx\), and if \(f(x)\leq0\) on \([m,n]\), then \(A=-\int_{m}^{n}f(x)dx\).

Step2: Analyze interval \([a,b]\)

On the interval \([a,b]\), the function \(y = f(x)\geq0\). So the area \(A_{1}=\int_{a}^{b}f(x)dx\).

Step3: Analyze interval \([0,c]\)

On \([0,c]\), the function \(y = f(x)\) has two parts. From \(0\) to some point \(x_0\in(0,c)\), \(f(x)\leq0\) and from \(x_0\) to \(c\), \(f(x)\geq0\). So the area \(A_{2}=-\int_{0}^{x_0}f(x)dx+\int_{x_0}^{c}f(x)dx\).

Step4: Analyze interval \([c,d]\)

On the interval \([c,d]\), \(f(x)\leq0\). So the area \(A_{3}=-\int_{c}^{d}f(x)dx\).

Step5: Analyze interval \([a,d]\)

Combining the above results, the area \(A_{4}=\int_{a}^{b}f(x)dx-\int_{b}^{x_0}f(x)dx+\int_{x_0}^{c}f(x)dx-\int_{c}^{d}f(x)dx\).

Answer:

(A) \(\int_{a}^{b}f(x)dx\)
(B) \(-\int_{0}^{x_0}f(x)dx+\int_{x_0}^{c}f(x)dx\)
(C) \(-\int_{c}^{d}f(x)dx\)
(D) \(\int_{a}^{b}f(x)dx-\int_{b}^{x_0}f(x)dx+\int_{x_0}^{c}f(x)dx-\int_{c}^{d}f(x)dx\)