Question

Use the concept of the definite integral to find the total area between the graph off(x) and thex-axis, by taking the limit of the associated right Riemann sum. Write the exact answer. Do not round. (Hint: Extra care isneeded on those intervals wheref(x) < 0. Remember that the definite integral represents a signed area.)f(x) = 4x2 - 16 on [-2, 2]

Answer 1

By the definition of definite integral, the integral of a function in a interval is the sum of the areas above the x-axis subtracted by the areas below the x-axis inside of the interval.

Given the function

`f(x)=4x^2-16`

Since our function is an upward parabola and our interval is [-2, 2], and those are the roots of our function, the whole region is under the x-axis, therefore, the area we want is minus the integral of our function on this interval.

`A=-\int ^2_(-2)(4x^2-16)dx`

Since we're dealing with an integral of a pair function in a symmetric interval, we can rewrite our integral as

`-\int ^2_(-2)(4x^2-16)dx=-2\int ^2_0(4x^2-16)dx`

Multiplying an integral by (-1) is the same as inverting the order of the limits.

`-2\int ^2_0(4x^2-16)dx=2\int ^0_2(4x^2-16)dx`

To find the area, we can just solve the integral

`\begin{gathered} A=2\int ^0_2(4x^2-16)dx=2\lbrack(4)/(3)x^3-16x\rbrack^0_2_{}^{} \\ A=2(32-(32)/(3)) \\ A=64-(64)/(3) \\ A=(128)/(3) \end{gathered}`

And this is our answer.