calculus integral question

Question

Consider the integral:

21

∫ x^3 dx

3



-Use all three methods to approximate this integral with n = 3

-Find the exact value of this integral

Answer 1

Left-hand rectangles: ∫₃²¹ x³ dx ≈ 6·3³ + 6·9³ + 6·15³

Right-hand rectangles: ∫₃²¹ x³ dx ≈ 6·9³ + 6·15³ + 6·21³

Trapezoids: Average of the above 2 answers

Exact value: ∫₃²¹ x³ dx = 1/4x⁴ ]₃²¹ = 1/4·21⁴ - 1/4·3⁴

Note: Left-hand approximation will be less than the exact value. Right-hand will be above the exact value. Trapezoids will also be above the exact value, but a closer approximation than the right-hand sum was.