the graph of $f(x)$ is shown below. evaluate $int_{-12}^{10} f(x),dx$ by interpreting it in terms of area.
the graph of $f(x)$ is shown below. evaluate $int_{-12}^{10} f(x),dx$ by interpreting it in terms of area.
Answer
Explanation:
Step1: Divide the region
The region under the curve from $x = - 12$ to $x=10$ can be divided into geometric - shapes (a triangle and a rectangle).
Step2: Calculate the area of the triangle
The base of the triangle from $x=-12$ to $x = - 5$ has length $b = 7$. The height of the triangle at $x=-5$ is $h = 10$. The area of a triangle is $A_{1}=\frac{1}{2}bh$. So $A_{1}=\frac{1}{2}\times7\times10 = 35$.
Step3: Calculate the area of the rectangle
The rectangle has length $l=10$ (from $x=-5$ to $x = 5$) and height $h = 10$. The area of a rectangle is $A_{2}=lh$. So $A_{2}=10\times10 = 100$.
Step4: Calculate the area of the second - triangle
The base of the triangle from $x = 5$ to $x=10$ has length $b = 5$. The height of the triangle at $x = 5$ is $h = 10$. The area of a triangle is $A_{3}=\frac{1}{2}bh$. So $A_{3}=\frac{1}{2}\times5\times10=25$.
Step5: Sum up the areas
The definite integral $\int_{-12}^{10}f(x)dx$ is equal to the sum of the areas of these geometric shapes. $\int_{-12}^{10}f(x)dx=A_{1}+A_{2}+A_{3}=35 + 100+25=160$.
Answer:
$160$