Question

evaluate the definite integral by interpreting it in terms of areas.
$$ int _ { 2 } ^ { 7 } ( 2 x - 10 ) d x $$

Explanation:

Step1: Find the x - intercept of the function \(y = 2x-10\)

Set \(y = 0\), then \(2x-10=0\), so \(x = 5\).

Step2: Split the integral based on the x - intercept

\(\int_{2}^{7}(2x - 10)dx=\int_{2}^{5}(2x - 10)dx+\int_{5}^{7}(2x - 10)dx\)
For \(y = 2x-10\), when \(x = 2\), \(y=2\times2 - 10=-6\); when \(x = 5\), \(y = 0\); when \(x = 7\), \(y=2\times7 - 10 = 4\).
The function \(y = 2x - 10\) is a straight line.
The integral \(\int_{a}^{b}f(x)dx\) can be interpreted as the net - area between the curve \(y = f(x)\), the \(x\) - axis, and the lines \(x=a\) and \(x = b\).
The area of a triangle is \(A=\frac{1}{2}\times base\times height\)
For \(\int_{2}^{5}(2x - 10)dx\): The base of the triangle is \(5 - 2=3\) and the height is \(| - 6|=6\). Since the function \(y = 2x-10\) is below the \(x\) - axis on the interval \([2,5]\), \(\int_{2}^{5}(2x - 10)dx=-\frac{1}{2}\times3\times6=-9\)
For \(\int_{5}^{7}(2x - 10)dx\): The base of the triangle is \(7 - 5 = 2\) and the height is \(4\). Since the function \(y = 2x-10\) is above the \(x\) - axis on the interval \([5,7]\), \(\int_{5}^{7}(2x - 10)dx=\frac{1}{2}\times2\times4 = 4\)

Step3: Calculate the value of the original integral

\(\int_{2}^{7}(2x - 10)dx=-9 + 4=-5\)

Answer:

\(-5\)