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

$\int ^2_{-2}14x^2-56dx$

Solution

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

Now

$\begin{gathered} \int ^2_{-2}14x^2dx-\int ^2_{-2}56dx \\ \frac{224}{3}-224 \\ \\ \frac{224}{3}-\frac{224}{1} \\ -\frac{448}{3} \end{gathered}$

The final answer is

$-\frac{448}{3}$