Chapter 5

In Exercises \(1-6,\) each \(c_{k}\) is chosen from the \(k\) th subinterval of a regular partition of the indicated interval into \(n\) subintervals of length \(\Delta x .\) Express the limit as a definite integral. $$\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \sqrt{4-c_{k}^{2}} \Delta x, \quad[0,1]$$

In Exercises 1-6, (a) use the Trapezoidal Rule with n = 4 to approximate the value of the integral. (b) Use the concavity of the function to predict whether the approximation is an overestimate or an underestimate. Finally, (c) find the integral's exact value to check your answer. $$\int_{0}^{4} \sqrt{x} d x$$

In Exercises \(1-20,\) find \(d y / d x\). $$y=\int_{2}^{x}\left(\tan ^{3} u\right) d u$$

Exercises \(5-8\) refer to the region \(R\) enclosed between the graph of the function \(y=2 x-x^{2}\) and the \(x\) -axis for \(0 \leq x \leq 2\) . (a) Sketch the region \(R\) . (b) Partition \([0,2]\) into 4 subintervals and show the four rectangles that LRAM uses to approximate the area of \(R .\) Repeat Exercise 1\((b)\) for RRAM and MRAM.

In Exercises \(1-6,\) each \(c_{k}\) is chosen from the \(k\) th subinterval of a regular partition of the indicated interval into \(n\) subintervals of length \(\Delta x .\) Express the limit as a definite integral. $$\lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left(\sin ^{3} c_{k}\right) \Delta x, \quad[-\pi, \pi]$$

In Exercises \(1-20,\) find \(d y / d x\). $$y=\int_{1}^{x} e^{u} \sec u d u$$

In Exercises 1-6, (a) use the Trapezoidal Rule with n = 4 to approximate the value of the integral. (b) Use the concavity of the function to predict whether the approximation is an overestimate or an underestimate. Finally, (c) find the integral's exact value to check your answer. $$\int_{0}^{\pi} \sin x d x$$

Show that the value of \(\int _ { 0 } ^ { 1 } \sin \left( x ^ { 2 } \right) d x\) cannot possibly be 2

In Exercises \(1-20,\) find \(d y / d x\). $$y=\int_{7}^{x} \frac{1+t}{1+t^{2}} d t$$

In Exercises \(7-12,\) evaluate the integral. $$\int_{-2}^{1} 5 d x$$