Question

I can't believe sometimes that English is my first language. What does this question mean?

"Here is a plot of f[x] = x^3 on [-2,2]

How does the plot give away the value of Integrate[x^3, {x, -2, 2}] ?​"

pictures coming ...

Answer 1

image attached

Answer 2

ok... so
we are given a function
\[\LARGE f(x) = x^3\] graphed between the intervals of -2 to 2

Answer 3

and then we are asked how the plot of that graph gave away the value when we are integrating from -2 to 2

Answer 4

you wouldn't mind if we integrate first right? do you know anti-derivatives?

Answer 5

Hint : if \(f(x)\) is an odd function,
\[\int\limits_{-a}^a f(x)\,dx = 0\]

Answer 6

I dont mind, but negative on the anti derivatives (I think)

Answer 7

when we are taking the antiderivative, we add one to the exponent and divide by the new exponent.

Answer 8

so since your exponent is 3. What's 3 +1 ?

Answer 9

so do they mean, I am to look at the plot, and then work out the value of the integration equation?

Answer 10

hmmm.. 5-1?

Answer 11

j/k 4

Answer 12

now evaluate
f(b)-f(a)
or f(2)-f(-2)

Answer 13

when the question says.. the value of ... do they mean that the equation itself is a value, or that the equation comes out to some value?

Answer 14

Notice that the graph is symmetric about origin, so f(x) is an odd function. You don't need to do any further work, the definite integral between a symmetric interval would be simply 0 because the positive area and negative area kill each other out.

Answer 15

oh, so by looking at the image.. I should see that it is symmetrical about zero, and therefore the sum of it's parts will be a negative area + an equal positive area = zero.

Answer 16

yes it is an odd function, so we do have symmetry around the origin . When we graph x^3 only we should have just