Question

In Exercises \(19-30,\) evaluate the integral using antiderivatives, as in Example \(4 .\) $$\int _ { 3 } ^ { 7 } 8 d x$$

Short answer

The value of the definite integral \(\int _ { 3 } ^ { 7 } 8 d x\) is \(32\).

Step-by-step solution

Find the Antiderivative of the Function

The function given in the problem is a constant, 8. The antiderivative of a constant function \(k\) is \(kx\), which means the antiderivative of our function is \(8x\).

Evaluate the Antiderivative at the Upper Bound

Place \(7\) in place of \(x\) in the function \(8x\) to get \(8\cdot7=56\). This is the result of the antiderivative function at the upper bound of \(7\).

Evaluate the Antiderivative at the Lower Bound

Similarly, place \(3\) in place of \(x\) in the function \(8x\) to get \(8\cdot3=24\). This is the result of the antiderivative function at the lower bound of \(3\).

Subtract the Lower Bound Result from the Upper Bound Result

In order to determine the value of the definite integral, subtract the lower bound result from the upper bound result: \(56-24=32\).