Question

refering to the figure, explain how to use definite integrals to find the area between the graph of ( y = f(x) ) and the ( x )-axis from ( x = a ) to ( x = d ).
choose the correct answer below.
a. integrate ( -f(x) ) over the interval ( a,d ), and then take the absolute value of the result.
b. integrate ( f(x) ) over the intervals ( a,b ) and ( c,d ), integrate ( -f(x) ) over the interval ( b,c ), and then sum the results.
c. integrate ( -f(x) ) over the intervals ( a,b ) and ( c,d ), integrate ( f(x) ) over the interval ( b,c ), and then sum the results.
d. integrate ( f(x) ) over the interval ( a,d ), and then take the absolute value of the result.

Explanation:

The area between a curve \(y = f(x)\) and the \(x -\)axis is calculated by integrating the function. When \(f(x)\geq0\) (above the \(x -\)axis), the integral \(\int_{a}^{b}f(x)dx\) gives the area. When \(f(x)\leq0\) (below the \(x -\)axis), the integral \(\int_{c}^{d}f(x)dx\) is negative. To get the area, we use \(\int_{c}^{d}-f(x)dx\) (since area is non - negative).
Looking at the intervals:

• For the interval \([a,b]\), the function \(y = f(x)\) is below the \(x -\)axis (\(f(x)\leq0\)), so the area is \(\int_{a}^{b}-f(x)dx\).
• For the interval \([b,c]\), the function \(y = f(x)\) is above the \(x -\)axis (\(f(x)\geq0\)), so the area is \(\int_{b}^{c}f(x)dx\).
• For the interval \([c,d]\), the function \(y = f(x)\) is below the \(x -\)axis (\(f(x)\leq0\)), so the area is \(\int_{c}^{d}-f(x)dx\).

Answer:

C. Integrate \(-f(x)\) over the intervals \([a,b]\) and \([c,d]\), integrate \(f(x)\) over the interval \([b,c]\), and then sum the results.