Math Problem Statement

Evaluate the expression involving definite integrals: (1/2) ∫(0 to 4) (8x - 2x^2) dx - (1/2) ∫(4 to 1/2) (8x - 2x^2) dx.

Solution

We are tasked with evaluating the given expression involving definite integrals:

12∫04(8x−2x2) dx−12∫412(8x−2x2) dx\frac{1}{2} \int_0^{4} (8x - 2x^2) \, dx - \frac{1}{2} \int_4^{\frac{1}{2}} (8x - 2x^2) \, dx

Let’s solve it step by step:

Step 1: Evaluate the first integral

∫04(8x−2x2) dx\int_0^{4} (8x - 2x^2) \, dx

  1. Find the antiderivative: ∫(8x−2x2) dx=4x2−23x3\int (8x - 2x^2) \, dx = 4x^2 - \frac{2}{3}x^3

  2. Apply the limits:

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Math Problem Analysis

Mathematical Concepts

Integral Calculus
Definite Integrals
Polynomial Integration

Formulas

∫(a to b) f(x) dx = F(b) - F(a), where F(x) is the antiderivative of f(x)
Antiderivative of ax^n = (a/(n+1))x^(n+1) + C

Theorems

Fundamental Theorem of Calculus

Suitable Grade Level

Grades 11-12 (Advanced High School Calculus)

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