Question

Use a definite integral to find the area under the curve between the given x-values.
f(x) = x−1 + x5 from x = 1 to x = 2

Answer 1

Question#18928

Use a definite integral to find the area under the curve between the given x-values.

$$f(x) = x - 1 + x5 \quad \text{from} \quad x = 1 \quad \text{to} \quad x = 2$$

Solution:

$$V = \int_{x_1}^{x_2} f(x) \, dx = \int_{1}^{2} (x - 1 + x^5) \, dx = \frac{1}{2} x^2 - x + \frac{1}{6} x^6 \Big|_{1}^{2} = \left(\frac{1}{2} 2^2 - 2 + \frac{1}{6} 2^6\right) - \left(\frac{1}{2} - 1 + 1\right) = 10\frac{1}{6}$$

Answer: $10\frac{1}{6}$ square units.