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 is needed on those intervals wheref(x) < 0. Remember that the definite integral represents a signed area.)

Answer 1

`\int ^2_(-2)14x^2-56dx`
`\begin{gathered} \int ^2_(-2)14x^2dx-\int ^2_(-2)56dx \\ \int ^2_(-2)14x^2dx=\lbrack(14(2)^3)/(3)\rbrack-\lbrack(14(-2)^3)/(3)\rbrack \\ \\ \\ =\lbrack\frac{14(8)^{}}{3}\rbrack-\lbrack\frac{14(-8)^{}}{3}\rbrack \\ =(112)/(3)+(112)/(3) \\ =(224)/(3) \\ \\ \\ \int ^2_(-2)56dx=56x \\ \int ^2_(-2)56dx=56(2)-56(-2) \\ \\ \int ^2_(-2)56dx=112+112 \\ \\ \int ^2_(-2)56dx=224 \end{gathered}`

Now

`\begin{gathered} \int ^2_(-2)14x^2dx-\int ^2_(-2)56dx \\ (224)/(3)-224 \\ \\ (224)/(3)-(224)/(1) \\ -(448)/(3) \end{gathered}`

The final answer is

`-(448)/(3)`