Evaluate each of the following definite integrals by using its geometric interpretation. (a) $\int_{7}^{12} 4 x-20 d x=$ (b) $\int_{3}^{10} 4 x-20 d x=$ Submit Answer

Step 1 ：The integrals represent the area under the curve of the function \( f(x) = 4x - 20 \) between the given limits of integration.

Step 2 ：The function is a straight line, so the area under the curve between any two points is a trapezoid.

Step 3 ：The area of a trapezoid is calculated using the formula \( A = \frac{1}{2}(b_1 + b_2)h \), where \( b_1 \) and \( b_2 \) are the lengths of the parallel sides and \( h \) is the height.

Step 4 ：For integral (a), the height \( h \) is the difference between the upper and lower limits of integration, which is \( 12 - 7 = 5 \).

Step 5 ：The lengths of the parallel sides for integral (a) are the values of the function at the limits, which are \( f(7) = 4(7) - 20 = 8 \) and \( f(12) = 4(12) - 20 = 28 \).

Step 6 ：Using the formula for the area of a trapezoid, the area under the curve for integral (a) is \( A = \frac{1}{2}(8 + 28) \cdot 5 = 90 \).

Step 7 ：For integral (b), the height \( h \) is \( 10 - 3 = 7 \).

Step 8 ：The lengths of the parallel sides for integral (b) are \( f(3) = 4(3) - 20 = -8 \) and \( f(10) = 4(10) - 20 = 20 \).

Step 9 ：Using the formula for the area of a trapezoid, the area under the curve for integral (b) is \( A = \frac{1}{2}(-8 + 20) \cdot 7 = 42 \).

Step 10 ：\( \boxed{90} \) is the final answer for integral (a).

Step 11 ：\( \boxed{42} \) is the final answer for integral (b).