Question

Corollary 3.11. Let $f \in \mathbb{R}$ on $[a, b]$ and $\phi: [a', b'] \to [a, b]$ be a strictly increasing, differentiable function. If $\phi' \in \mathbb{R}$ on $[a', b']$, then $\int_a^b f(x)dx = \int_{a'}^{b'} f(\phi(y))\phi'(y)dy$. Problem 3.12. Prove Corollary 3.11

          Corollary 3.11. Let $f \in \mathbb{R}$ on $[a, b]$ and $\phi: [a', b'] \to [a, b]$ be a strictly increasing, differentiable function. If $\phi' \in \mathbb{R}$ on $[a', b']$, then
$\int_a^b f(x)dx = \int_{a'}^{b'} f(\phi(y))\phi'(y)dy$.
Problem 3.12. Prove Corollary 3.11

Corollary 3.11. Let f ∈ℝ on [a, b] and ϕ: [a', b'] → [a, b] be a strictly increasing, differentiable function. If ϕ' ∈ℝ on [a', b'], then
^b f(x)dx = ∫a'^b' f(ϕ(y))ϕ'(y)dy.
Problem 3.12. Prove Corollary 3.11

Added by Geoffrey H.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

07/26/2022

Step 1

We are also given that ' is a function in R on [a',b']. Show more…

Show all steps

Thanks for your feedback!

Corollary 3.11. Let f e R on [a,b] and :[a',b'] -> [a,b] be a strictly increas- ing, differentiable function. If ' e R on [a',b'], then f((y))'(y)dy Problem 3.12. Prove Corollary 3.11