Calculus True or False question

Question

Consider the linear function f(x)=2x+6 and the region bounded by its graph and the x-axis on the interval [2,6]. Suppose the area of this region is approximated using midpoint Riemman sums. Then the approximations give the exact area of the region for any number of subintervals. True or false. if false, give an explanation of counterexample.

Answer 1

Since the function is a straight line, then the triangle formed between the midpoint and the left hand side and the triangle formed between the midpoint and the righthand side will have the same height.

Since we are using the midpoint of the rectangle, the base of both triangles will be the same as well.

This means that both will have the same area. Area of a triangle is 1/2 * base * height.

On the left, since the slope of the function is 2 and is positive, the Riemman sum will over estimate by the area of the triangle, and on the right hand side, the Riemman sum will underestimate the area.

These 2 errors will cancel out and give an exact representation of the area under the curve.

So the answer is true.

Check: Pick the point x = 4 and the size of the rectangle = 1. Then to figure out the height of the triangle at x = 4, y = 2×8 + 6 = 14 and on the right side, x = 4.5 so y = 2×4.5 + 6 = 15 so that the height of the triangle is 1.

Area is 1/2 × base × height = 1/2 ×1/2× 1 = 1/4. So we have an underestimation of 1/4 on the right.

On the left, y = 14 at the midpoint as before, but at the left of the triangle, y = 2 × 3.5 + 6 = 13 so that the height of that triangle is also 1 ie 14 - 13.

So then the area of the overestimation is 1/2 × 1/2 × 1 = 1/4.

Both triangles are equal, and the errors cancel out.

Answer 2

Was just thinking about a similar question just a few minutes ago. The answer is true and it's true for any function that is a line. You can see this by drawing out a the line and the rectangles. The area under the line and outside the rectangles is the same as the area over the line and inside the rectangles, making the area under the line and the sum of the areas of the rectangles the same.