Consider the curve f(x) = 4 - x^2 (Riemann sum and definite integrals)

Question

1) Write (without evaluating) a definite integral that gives the exact signed area under the curve between x = -2 and x = 1.

2) Write (without evaluating) an expression for the exact area under the curve between x = -2 and x = 1 as the limit of a Riemann sum.

3) Estimate the area under the curve f between x = -2 and x = 1 using a left Riemann sum with three rectangles (draw a diagram is possible).

4) Is your estimate in 3) an overestimate or an underestimate? Please justify your answer.

Answer 1

1) Area A = ∫_-2¹(4 - x²)dx

2) Area A = lim 3/n·∑_k=1ⁿ (4 - (- 2 + 3k/n)²) Here is right sum

n→∞

3) Area A ≈ L₃ = 3/3∑_k=0²[4 - (- 2 + 3k/3)²] = 1(0 + 3 + 4) = 7

4) We have underestimate because function is even and it increasing from - 2 to 0 and decreasing only from0 to1.