Question

Determine the area under the curve $y=x^3-x^2$ from $x=1$ to $x=5$.

• A. 344/3 sq unit
• B. 342/3 sq unit
• C. 345 sq unit
• D. 344 sq unit

Answer 1

Answer: (A)

344/3 sq unit

Answer 2

The area under the curve $y=f(x)$ from $x=a$ to $x=b$ is calculated by the definite integral $\int_{a}^{b} f(x) dx$. For $y=x^3-x^2$ from $x=1$ to $x=5$, we compute $\int_{1}^{5} (x^3 - x^2) dx$.

1. Find the antiderivative: $\int (x^3 - x^2) dx = \frac{x^4}{4} - \frac{x^3}{3}$.

2. Evaluate the antiderivative at the limits: $\left[ \frac{x^4}{4} - \frac{x^3}{3} \right]_{1}^{5} = \left( \frac{5^4}{4} - \frac{5^3}{3} \right) - \left( \frac{1^4}{4} - \frac{1^3}{3} \right)$.

3. Calculate the values: $\left( \frac{625}{4} - \frac{125}{3} \right) - \left( \frac{1}{4} - \frac{1}{3} \right)$.

4. Simplify fractions: $\left( \frac{1875 - 500}{12} \right) - \left( \frac{3 - 4}{12} \right) = \frac{1375}{12} - \left( \frac{-1}{12} \right)$.

5. Perform subtraction: $\frac{1375}{12} + \frac{1}{12} = \frac{1376}{12}$.

6. Reduce the fraction: $\frac{1376}{12} = \frac{344}{3}$.

Therefore, the area under the curve is $\frac{344}{3}$ sq unit.