Question

In the following exercises, use a calculator to estimate the area under the curve by computing T₁₀ , the average of the left- and right-endpoint Riemann sums using N=10 rectangles. Then, using the Fundamental Theorem of Calculus, Part 2, determine the exact area.

164. [T]y=x² over [0,4]

Answer 1

To estimate and find the exact area under the curve y = x^2 over the interval [0,4], we can use the left-endpoint Riemann sum with N=10 rectangles and the Fundamental Theorem of Calculus, Part 2.

Step-by-step explanation:

To estimate the area under the curve y = x^2 over the interval [0,4] using the left-endpoint Riemann sum with N=10 rectangles, we divide the interval into 10 equal subintervals. The width of each rectangle is 4/10 = 0.4.

Next, we evaluate the function at the left endpoint of each subinterval and multiply it by the width to find the areas of the rectangles. Adding up these areas gives us an estimate of the area under the curve.

Using the Fundamental Theorem of Calculus, Part 2, we can find the exact area under the curve by evaluating the antiderivative of the function over the interval [0,4].