the graph of $f(x)$ is shown below. evaluate $int_{-11}^{9} f(x) dx$ by interpreting it in terms of area.
the graph of $f(x)$ is shown below. evaluate $int_{-11}^{9} f(x) dx$ by interpreting it in terms of area.
Answer
Explanation:
Step1: Divide the region under the curve
The region between (x = - 11) and (x=9) under (y = f(x)) can be divided into a triangle on the left - hand side, a rectangle in the middle, and a triangle on the right - hand side.
Step2: Calculate the area of the left - hand triangle
The base of the left - hand triangle is (b_1=5) (from (x=-11) to (x = - 6)) and the height (h_1 = 20). The area of a triangle is (A=\frac{1}{2}bh), so (A_1=\frac{1}{2}\times5\times20 = 50).
Step3: Calculate the area of the rectangle
The rectangle has length (l = 10) (from (x=-6) to (x = 4)) and width (w = 20). The area of a rectangle is (A=lw), so (A_2=10\times20=200).
Step4: Calculate the area of the right - hand triangle
The base of the right - hand triangle is (b_2 = 5) (from (x = 4) to (x=9)) and the height (h_2=10). The area of a triangle is (A=\frac{1}{2}bh), so (A_3=\frac{1}{2}\times5\times10 = 25).
Step5: Calculate the definite integral
The definite integral (\int_{-11}^{9}f(x)dx) is equal to the sum of the areas of these three regions. So (\int_{-11}^{9}f(x)dx=A_1 + A_2+A_3=50 + 200+25=275).
Answer:
275