Calculus/Fundamental Theorem of Calculus/Solutions

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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.