Question

question 5 (1 point)
the (bounded) area between the curves ( f ( x ) = 15 x ^ { 4 } ) and ( g ( x ) = 15 x ^ { 2 } ) is
. (your answer should be a number.)
view hint for question 5

Explanation:

Step1: Find intersection points

Set $15x^4 = 15x^2$ → $x^4 - x^2 = 0$ → $x^2(x^2 - 1)=0$ → $x=-1,0,1$.

Step2: Determine upper/lower functions

For $x \in (-1,0)\cup(0,1)$, $15x^2 > 15x^4$; symmetry about y-axis, so compute 2×area from 0 to1.

Step3: Integrate difference from 0 to1

Area = $2\int_{0}^{1}(15x^2 -15x^4)dx = 2×15\int_{0}^{1}(x^2 -x^4)dx$

Step4: Evaluate integral

$\int_{0}^{1}(x^2 -x^4)dx = [\frac{x^3}{3}-\frac{x^5}{5}]_0^1 = \frac{1}{3}-\frac{1}{5} = \frac{2}{15}$

Step5: Compute total area

$2×15×\frac{2}{15}=4$

Answer:

4