Section 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.
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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.
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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.
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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