How do you evaluate the definite integral by the limit definition given #int 3x^2dx# from [1,3]?

Question

Isabella Ackerman · Accepted Answer

26

#int_1^3(3x^2)dx# #=[(cancel(3)x^3)/cancel(3)]_1^3# #[x^3]^3-[x^3]_1# #=3^3-1^3# #=27-1=26#

Christopher Allers · Answer

undefined To evaluate the definite integral by the limit definition for ∫(1 to 3) 3x^2 dx, follow these steps:

1. Start with the limit definition of the definite integral: ∫(a to b) f(x) dx = lim(n→∞) Σ[f(xi) Δx], where Δx = (b - a) / n, and xi is a point in the ith subinterval.

2. Determine the function f(x) in the integral, which is 3x^2.

3. Calculate Δx: Δx = (3 - 1) / n = 2/n.

4. Choose the points xi within each subinterval. Since the interval [1, 3] is divided into n equal subintervals, xi can be chosen as follows: xi = 1 + iΔx, where i ranges from 0 to n-1.

5. Express f(xi) in terms of xi: f(xi) = 3(xi)^2 = 3(1 + iΔx)^2.

6. Rewrite the integral using the limit definition and the expressions for Δx and f(xi): ∫(1 to 3) 3x^2 dx = lim(n→∞) Σ[3(1 + iΔx)^2 Δx].

7. Expand (1 + iΔx)^2 and simplify the expression inside the summation.

8. Evaluate the summation Σ[3(1 + iΔx)^2 Δx].

9. Take the limit of the summation as n approaches infinity to find the value of the definite integral.

10. Once you have the limit, evaluate it to get the numerical value of the definite integral.

This process involves algebraic manipulation and taking limits, resulting in the evaluation of the definite integral by the limit definition.