Question

Find area bounded by the inverse of bijective function f(x)=4x^3+6x the x axis and ordinate x=0 and x=44

Answer 1

Answer:

The area bounded bu the inverse of bijective function $f(x) = 4x^3+6x$, the axis $x$ and the coordinates $x = 0$ and $x = 44$ is $60$.

Step-by-step explanation:

We will denote the inverse function of $f(x)$ by $f^{-1}(y)$. That is, we will solve the integral of f^{-1}(y) from y = 0 to y = 44.

If $f(a) = c$ e $f(b) = d$, we have the formula

$\int_c^df^{-1}(y) dy + \int_a^bf(x)dx = bd -ac$

Since $f(0) = 0$ and $f(2) = 44$, we get

$\int_0^44f^{-1}(y) dy =- \int_0^2f(x)dx + 2\cdot44-0\cdot 0$

But we know $\int 4x^3+6x = x^4+3x^2$

So $\int_0^2f(x)dx  = 16+12 = 28$

Therefore

$\int_0^44f^{-1}(y) dy =- 28+ 2\cdot44 = 60$