Estimate the area under the curve f(x) = 16 – x^2 from x = 0 to x = 3 by using three inscribed (under the curve) rectangles. Answer to the nearest integer.
39
To estimate the area under the curve using inscribed rectangles, we can use the following steps:
1. Divide the range of integration (in this case, from x = 0 to x = 3) into equal intervals. Since we have three rectangles, we divide the range into three equal intervals: [0, 1], [1, 2], [2, 3].
2. Calculate the width of each rectangle by finding the difference between the endpoints of each interval. In this case, the width of each rectangle is 1 unit.
3. Determine the height of each rectangle by evaluating the function f(x) = 16 – x^2 at the left endpoint of each interval, since we are using inscribed rectangles. Evaluate f(0), f(1), and f(2) to find the heights of the three rectangles.
4. Calculate the area of each rectangle by multiplying its width by its height.
5. Sum up the areas of the three rectangles to get an estimate of the total area under the curve.
Let's follow these steps:
1. Divide the range [0, 3] into three equal intervals: [0, 1], [1, 2], [2, 3].
2. The width of each rectangle is 1 unit.
3. Evaluate the function at the left endpoint of each interval:
At x = 0: f(0) = 16 - 0^2 = 16.
At x = 1: f(1) = 16 - 1^2 = 15.
At x = 2: f(2) = 16 - 2^2 = 12.
4. Calculate the area of each rectangle:
Rectangle 1: Area = width * height = 1 * 16 = 16.
Rectangle 2: Area = width * height = 1 * 15 = 15.
Rectangle 3: Area = width * height = 1 * 12 = 12.
5. Sum up the areas of the three rectangles:
Total estimated area = 16 + 15 + 12 = 43.
Therefore, the estimated area under the curve f(x) = 16 – x^2 from x = 0 to x = 3, using three inscribed rectangles, is approximately 43 square units.
16-x^2 is concave down, so you want to use the right endpoints. Each interval has width 1, so you just have
2(f(1)+f(2)+f(3)) = 2(15+12+7) = 68
There are several good online Riemann Sum calculators. You can use them to verify your work.