Sction 5.1: Areas 

1. Divide the interval $[-2,3]$ into five subintervals of equal length. Then approximate the area under the curve $y=f(x)$ on $[-2,3]$ sketching the corresponding rectangles using left endpoints of each subinterval. Repeat this using right end points of each subinterval.

solution

2. Estimate the area under the graph of $f(x) = 8 \cos(x)$ from $x = 0$ to $x= \pi / 2$ using four approximating rectangles and right endpoints. Repeat this using left endpoints of the approximating rectangles.

solution

3. Divide the interval $[0,4]$ into four subintervals of equal length. Then approximate the area under the curve $f(x)=\sqrt{x}$ on $[0, 4]$ sketching the corresponding rectangles using (a) right-end points, and (b) mid-points of each subinterval.

solution

4. Consider the definite integral $\int_0^3 (x^2+1) \, dx$. Approximate the integral using a Riemann sum with $n=3$ using left endpoints.    solution

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