Help on calculus please: Area approximation

Question

Please estimate the area under the curve f(x) = x² from x = 1 to x = 5 by using four inscribed (under the curve) rectangles. Can you also give the answer to the nearest integer, thank you!

Answer 1

First, we must break the interval x = 1 to x = 5 into four sections in order to approximate with 4 rectangles.

We use 4 equal sections (which works nicely because the interval is 4 units wide).

For each rectangle, we must find a height, we can use either the left or right endpoint of the section to find the height. However, the problem asks for the inscribed (under the curve) rectangles, so we must use the left endpoints. Using the right endpoints will give us rectangles that are above the curve because the function f(x) = x^2 is strictly increasing.

The first section is from x = 1 to x = 2. Using the left endpoints, the height of the rectangle will be the height of the function at x = 1. f(1) = 1^2 = 1. So our first rectangle has a width of 1 unit and a height of 1 units. The first rectangle has an area 1*1 = 1.

The next section is from x = 2 to x = 3. We must continue to use the left endpoints, so the height is f(2) = 2^2 = 4. The width is still 1 unit, so the area of the next rectangle is 4.

The third section is from x = 3 to x = 4. The height is f(3) = 3^2 = 9. The width is still 1 unit, so the area of the next rectangle is 9.

The final section is from x = 4 to x = 5. The height is f(4) = 4^2 = 16. The width is still 1 unit, so the area of the next rectangle is 16.

Finally, we add the areas of each rectangle to calculate the Reimann Sum. 1+4+9+16 = 30 square units.