Question

evaluate the integral by interpreting it in terms of areas.

$$ int _ { - 7 } ^ { 8 } ( 10 - 5 x ) d x $$

$$ - \frac { 295 } { 2 } $$

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Explanation:

Step1: Find the function values at endpoints

When \(x = 7\), \(y_1=10 - 5\times7=10 - 35=- 25\).
When \(x = 8\), \(y_2=10 - 5\times8=10 - 40=-30\).

Step2: Use the formula for the area of a trapezoid

The integral \(\int_{a}^{b}f(x)dx\) (where \(f(x)\) is a linear function) can be interpreted as the area of a trapezoid. The formula for the area of a trapezoid is \(A=\frac{(y_1 + y_2)(b - a)}{2}\). Here, \(a = 7\), \(b = 8\), \(y_1=-25\), \(y_2=-30\).
Substitute into the formula: \(A=\frac{(-25-30)(8 - 7)}{2}\).

Step3: Calculate the value

First, simplify the numerator: \(-25-30=-55\), and \(8 - 7 = 1\). Then \(A=\frac{-55\times1}{2}=-\frac{55}{2}\).

Answer:

\(-\frac{55}{2}\)