Answer
The given integral is:
$$
\int_{0}^{1}(\sqrt[3]{1-x^{7}}-\sqrt[7]{1-x^{3}}) d x=0
$$
Work Step by Step
Evaluate
$$
\int_{0}^{1} (\sqrt[3] {1-x^{7 }} - \sqrt[7] {1-x^{3 }} )d x
$$
The given integral represents the difference of the shaded areas, which appears to be 0. It can be calculated by integrating with respect to either $x$ or $y$ so we find $x$ in terms of $y$ for each curve:
$$
y=\sqrt[3]{1-x^{7}} \Rightarrow x=\sqrt[7]{1-y^{3}}
$$
$$
y=\sqrt[7]{1-x^{3}} \Rightarrow x=\sqrt[3]{1-y^{7}},
$$
so
$$
\int_{0}^{1}(\sqrt[3]{1-y^{7}}-\sqrt[7]{1-y^{3}}) d y=\int_{0}^{1}(\sqrt[7]{1-x^{3}}-\sqrt[3]{1-x^{7}}) d x
$$
But this equation is of the form $ z=-z. $ So
$$
\int_{0}^{1}(\sqrt[3]{1-x^{7}}-\sqrt[7]{1-x^{3}}) d x=0
$$
