Question

Which of the following integrals represents the area of the region bounded by x = 0, x = 1 and the functions f(x) = -2x4 and g(x) = x2?
the integral from 0 to 1 of the quantity, x squared plus 2 times x to the 4th power, dx

the integral from 0 to 1 of the quantity, x squared minus 2 times x to the 4th power, dx

the integral from 0 to 1 of the quantity, negative 2 times x to the 4th power minus x squared, dx

the integral from 0 to 1 of the quantity, x to the 4th power minus 4 times x to the 8th power, dx

Answer 1

Answer:

C. $\int\limits^0_1 {-2x^4-x^2} \, dx$ is correct

Step-by-step explanation:

We are given that,

The region bounded by $x= 0, x=1, f(x)= -2x^4, g(x)=x^2$ is given in the figure below.

Now, as we have,

Area of the bounded region = $\int\limits^a_b {f(x)-g(x)} \, dx$,

where f(x) represents the upper curve and f(x) represents the lower curve in the bounded region.

So, as we see that,

The upper curve in the given region is $f(x)= -2x^4$ and the lower curve is $g(x)=x^2$.

Thus, the integral showing the area of the given region is,

$\int\limits^0_1 {-2x^4-x^2} \, dx$

Hence, option C is correct.