# 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.
$\int _{0}^{1}x^{6}dx={\frac {x^{7}}{7}}{\biggr |}_{0}^{1}={\frac {1^{7}}{7}}-{\frac {0^{7}}{7}}=\mathbf {{\frac {1}{7}}=0.{\overline {142857}}}$ 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.
$\int _{1}^{2}x^{6}dx={\frac {x^{7}}{7}}{\biggr |}_{1}^{2}={\frac {2^{7}}{7}}-{\frac {1^{7}}{7}}={\frac {128}{7}}-{\frac {1}{7}}=\mathbf {{\frac {127}{7}}=18.{\overline {142857}}}$ 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.
$\int _{0}^{2}x^{6}dx={\frac {x^{7}}{7}}{\biggr |}_{0}^{2}={\frac {2^{7}}{7}}-{\frac {0^{7}}{7}}=\mathbf {{\frac {128}{7}}=18.{\overline {285714}}}$ This is consistent with the bounds we calculated in exercise 4 in section 4.1.