Question

Please help me with my calculus homework, only question 3****

Answer 1

I would start by stating the Fundamental Theorem of Calculus which states that;

If a function f is continuous on a closed interval [a,b] and F is an antiderivative of f on the interval [a,b], then

`\int ^b_af(x)dx=F(b)-F(a)\text{ }`

Let

`\begin{gathered} f(x)=x^3-6x^{} \\ F(x)=\int f(x)dx=\int (x^3-6x)dx \end{gathered}`

Recall that;

`\int x^n=(x^(n+1))/(n+1),n\\e-1`

That implies that,

`F(x)=\int (x^3-6x)dx=\int x^3dx-\int 6xdx=(x^4)/(4)-6((x^2)/(2))=(x^4)/(4)-3x^2+C`

Applying the Fundamental Theorem of Calculus, where a=0, b=3

`\begin{gathered} \int ^3_0(x^3-6x)dx=F(3)-F(0) \\ F(3)=(3^4)/(4)-3(3)^2+C=(81)/(4)-27+C=-(27)/(4)+C \\ F(0)=(0^4)/(4)-3(0)^2+C=C \\ \Rightarrow\int ^3_0(x^3-6x)dx=-(27)/(4)+C-C=-(27)/(4) \end{gathered}`

So the answer is -27/4