# Calculus/Fundamental Theorem of Calculus/Solutions

1. Evaluate $\int_0^1 x^6 dx$. Compare your answer to the answer you got for exercise 1 in section 4.1.

$\int_0^1 x^6 dx = \frac{x^7}{7}\biggr|_0^1 = \frac{1^7}{7}-\frac{0^7}{7} = \mathbf{\frac{1}{7} = 0.\overline{142857}}$
This is consistent with the bounds we calculated in exercise 1 in section 4.1.

2. Evaluate $\int_1^2 x^6 dx$. Compare your answer to the answer you got for exercise 2 in section 4.1.

$\int_1^2 x^6 dx = \frac{x^7}{7}\biggr|_1^2 = \frac{2^7}{7}-\frac{1^7}{7} = \frac{128}{7}-\frac{1}{7} = \mathbf{\frac{127}{7} = 18.\overline{142857}}$
This is consistent with the bounds we calculated in exercise 2 in section 4.1.

3. Evaluate $\int_0^2 x^6 dx$. Compare your answer to the answer you got for exercise 4 in section 4.1.

$\int_0^2 x^6 dx = \frac{x^7}{7}\biggr|_0^2 = \frac{2^7}{7}-\frac{0^7}{7} = \mathbf{\frac{128}{7} = 18.\overline{285714}}$
This is consistent with the bounds we calculated in exercise 4 in section 4.1.