the piece - wise function f(x) is graphed below. use geometric formulas to evaluate the following definite…
the piece - wise function f(x) is graphed below. use geometric formulas to evaluate the following definite integral. ∫₀¹⁰ f(x) dx
Answer
Explanation:
Step1: Divide the region
Divide the region under the curve from $x = 0$ to $x=10$ into geometric - shapes. We have two triangles and a rectangle.
Step2: Calculate the area of the first triangle
The first triangle has base $b_1 = 4$ and height $h_1=- 5$. The area of a triangle is $A=\frac{1}{2}bh$. So, $A_1=\frac{1}{2}\times4\times(-5)=- 10$.
Step3: Calculate the area of the second triangle
The second triangle has base $b_2 = 2$ and height $h_2 = 1$. So, $A_2=\frac{1}{2}\times2\times1 = 1$.
Step4: Calculate the area of the rectangle
The rectangle has length $l = 4$ and width $w = 1$. The area of a rectangle is $A=lw$. So, $A_3=4\times1 = 4$.
Step5: Calculate the definite - integral
The definite integral $\int_{0}^{10}f(x)dx$ is the sum of the areas of these geometric shapes. $\int_{0}^{10}f(x)dx=A_1 + A_2+A_3=-10 + 1+4=-5$.
Answer:
$-5$