Evaluate the following limits of Riemann sums on the indicated intervals.
Recall:
On
\( [-2,20]\), \(\lim _{n\to \infty }\sum _{k=1}^n x^* \Delta x=\int _{-2}^{20}x \d x\). This integral gives you the net area of the region between the curve \(y=x\) and the
\(x-\)axis.
![[Picture]](definiteIntegral5.online-13288629838176ec6450ac295b626f2c.svg)
Express the
limit as a definite integral. Then use properties of definite integrals, symmetry and
geometry.
\begin{align*} \text {On } [-2,20],\,\lim _{n\to \infty }\sum _{k=1}^n x^* \Delta x &= \answer {198}\\ \text {On } [-2,2],\,\lim _{n\to \infty }\sum _{k=1}^n 6(x^*)^3 \Delta x &= \answer {0}\\ \text {On } [-7,7],\,\lim _{n\to \infty }\sum _{k=1}^n 6\sin (x^*) \Delta x &= \answer {0}\\ \text {On } [-7,7],\,\lim _{n\to \infty }\sum _{k=1}^n 6(x^*)^2\sin (x^*) \Delta x &= \answer {0}\\ \text {On } [4,7],\,\lim _{n\to \infty }\sum _{k=1}^n \sqrt {9-(x^*-4)^2} \Delta x &= \answer {\frac {9\pi }{4}} \end{align*}