Evaluate each of the following definite integrals by using its geometric interpretation.
(a) ${\int }_7^{12}4x-20dx=$
(b) ${\int }_3^{10}4x-20dx=$
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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 ：$\boxed{{\displaystyle 90}}$ is the final answer for integral (a).

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