Math / Calculus

QuestionGraph the integrand. $\begin{array}{c} \int_{-4}^{1} x^{3} d x \\ f(x)= \end{array}$ $\square$

Studdy Solution

STEP 1

What is this asking? We need to draw the graph of $x^3$ and shade the area underneath it between $x = -4$ and $x = 1$ . Watch out! The graph of $x^3$ can be tricky!
Remember that it's negative for negative $x$ values and passes through $(0, 0)$ .

STEP 2

1. Draw the graph
2. Shade the area

STEP 3

Alright, let's get visual!
We're dealing with the function $f(x) = x^3$ .
This is a cubic function, so it's going to have a cool curvy shape!

STEP 4

Let's pick some key points to help us draw this graph accurately.
When $x = 0$ , $f(x) = 0^3 = 0$ , so we have the point $(0, 0)$ .
When $x = 1$ , $f(x) = 1^3 = 1$ , giving us $(1, 1)$ .
When $x = -1$ , $f(x) = (-1)^3 = -1$ , giving us $(-1, -1)$ .
Let's also consider $x = 2$ , where $f(x) = 2^3 = 8$ , so $(2, 8)$ , and $x = -2$ , where $f(x) = (-2)^3 = -8$ , so $(-2, -8)$ .

STEP 5

Now, connect the dots smoothly!
Remember that cubic functions have that characteristic S-shape.
It comes up from the bottom left, passes through the origin, and then curves upwards to the right.
It's like a graceful swoop!

STEP 6

The problem asks us to find the definite integral from $x = -4$ to $x = 1$ .
This means we need to shade the area between the curve of $x^3$ , the x-axis, and the vertical lines at $x = -4$ and $x = 1$ .

STEP 7

First, draw a vertical line at $x = -4$ .
Since $(-4)^3 = -64$ , this line goes all the way down to $(-4, -64)$ .
Then, draw another vertical line at $x = 1$ .
This one goes to $(1, 1)$ .

STEP 8

Now, shade the region enclosed by these two vertical lines, the x-axis, and the curve of $f(x) = x^3$ .
Notice that the area to the left of the origin (between $x = -4$ and $x = 0$ ) is below the x-axis, so we shade it in.
The area to the right of the origin (between $x = 0$ and $x = 1$ ) is above the x-axis, so we shade that in too.
We now have two shaded regions, one below the x-axis and one above it!

STEP 9

The solution is the graph of $f(x) = x^3$ with the area between the curve and the x-axis shaded from $x = -4$ to $x = 1$ .
Remember that the shaded region to the left of the origin represents a negative area, while the shaded region to the right represents a positive area!

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