Integral Calculus
Area using inverse function; area switching formula
MJMT_Full_Test_05
Grade 12
Question:
Area enclosed by $y=g(x)$, $x=1$ and $x=37$, where $g(x)$ is the inverse of $f(x)=x^3+3x+1$, is
$\dfrac{91}{4}$
$\dfrac{297}{4}$
$\dfrac{207}{4}$
None of these
Step-by-Step Solution
Key Concept: Use the inverse function area formula: $\int_a^b g(x)\,dx = b\cdot f^{-1}(b) - a\cdot f^{-1}(a) - \int_{f^{-1}(a)}^{f^{-1}(b)} f(t)\,dt$. Note $f(0)=1$ and $f(3)=37$.
Area $= 37\cdot3 - \int_0^3(x^3+3x+1)\,dx = 111 - \frac{147}{4} = \frac{297}{4}$.
Correct Answer: 2