Math / Calculus

Questionoblem $1 \vee$ oblem $2 \checkmark$ oblem $3 \checkmark$ oblem 4 blem 5 blem $6 \checkmark$ blem $7 \checkmark$ blem 8 lem 9

Studdy Solution

STEP 1

What is this asking? We need to find the area under the line $5x - 1$ from $x = 0$ to $x = 5$ . Watch out! Don't forget that area below the x-axis counts as negative!

STEP 2

1. Find the antiderivative.
2. Evaluate the antiderivative at the boundaries.
3. Calculate the definite integral.

STEP 3

Let's find the antiderivative of our function $5x - 1$ .
Remember, the power rule for integration says that the antiderivative of $x^n$ is $\frac{x^{n+1}}{n+1}$ .

STEP 4

The antiderivative of $5x$ (which is the same as $5x^1$ ) is $5 \cdot \frac{x^{1+1}}{1+1} = \frac{5x^2}{2}$ .
We multiplied by $x$ raised to the old power plus one, and then divided by the new power!

STEP 5

The antiderivative of $-1$ (which is the same as $-1x^0$ ) is $-1 \cdot \frac{x^{0+1}}{0+1} = -x$ .

STEP 6

So, the complete antiderivative of $5x - 1$ is $\frac{5x^2}{2} - x$ .
Don't forget the constant of integration, which we can write as $+ C$ .
Since we're doing a definite integral, the $+C$ will disappear later, but it's good practice to include it!

STEP 7

Now, we evaluate our antiderivative at the upper limit of integration, which is $x = 5$ .
Substituting $x = 5$ into $\frac{5x^2}{2} - x + C$ , we get $\frac{5 \cdot 5^2}{2} - 5 + C = \frac{125}{2} - 5 + C = \frac{125}{2} - \frac{10}{2} + C = \frac{115}{2} + C$ .

STEP 8

Next, we evaluate at the lower limit of integration, $x = 0$ .
Substituting $x = 0$ into $\frac{5x^2}{2} - x + C$ , we get $\frac{5 \cdot 0^2}{2} - 0 + C = 0 + C = C$ .

STEP 9

To find the definite integral, we subtract the value of the antiderivative at the lower limit from the value at the upper limit.

STEP 10

So, we have $(\frac{115}{2} + C) - (C) = \frac{115}{2}$ .
See how the $+C$ and $-C$ add to zero?
This always happens with definite integrals!

STEP 11

The final answer, the definite integral of $5x - 1$ from $0$ to $5$ , is $\frac{115}{2}$ , or $57.5$ !

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