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I can't see what to do in the two STEP level questions below. Can anyone give me a clue, please?

The function $f$ satisfies the condition $f'(x) > 0$ for $a \leq x \leq b$, and $g$ is the inverse of $f$.

  1. By making a suitable change of variable, prove that $$\int_{a}^{b} f(x) \, dx = b\beta - a\alpha - \int_\alpha^\beta g(y) \, dy$$ where $\alpha = f(a)$ and $\beta = f(b)$. Interpret this formula geometrically, in the case where $\alpha$ and $a$ are both positive.

  2. Prove similarly and interpret (for $\alpha > 0$ and $a > 0$) the formula $$2\pi\int_{a}^{b} xf(x) \, dx = \pi(b^2\beta - a^2\alpha) - \pi\int_\alpha^\beta [g(y)]^2 \, dy.$$

This is the simplest situation I can come up with but something does not add up.

enter image description here

gary
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    Welcome to [math.se] SE. Take a [tour]. You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an [edit]): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance. – Martin R Nov 26 '23 at 08:26
  • For the first part, see, e.g., https://math.stackexchange.com/q/155493/42969 – found with Approach0 – Martin R Nov 26 '23 at 08:29
  • Please show some work, and ask only one question per post. – Anne Bauval Nov 26 '23 at 08:29
  • In the $n$th problem, evaluate $\int x^{n-1}f(x)dx$ with integration by parts using $u=f(x),,v=x^n/n$, then substitution using $y=f(x)$. To interpret the results, I recommend drawing diagrams. – J.G. Nov 26 '23 at 08:59
  • Your graphic interpretation of $\int_\alpha^\beta g(y) , dy$ is wrong: it should be bounded by the same curve $y=f(x)$ as the other integral. https://en.wikipedia.org/wiki/Integral_of_inverse_functions – Anne Bauval Nov 26 '23 at 22:32

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