We are tasked with finding the values of $b$ such that

$\int_{-3}^b |x| \, dx = 45.$

Step 1: Break down the absolute value function

The absolute value function $|x|$ behaves differently depending on whether $x$ is positive or negative. Thus, we need to split the integral based on the value of $b$.

• If $b \geq 0$, the absolute value function simplifies to $|x| = x$ for $x \geq 0$, and $|x| = -x$ for $x < 0$.
• If $b < 0$, we have $|x| = -x$ for all $x$ in the range $[-3, b]$.

Step 2: Case 1 — $b \geq 0$

For $b \geq 0$, we split the integral at $x = 0$. We have:

$\int_{-3}^b |x| \, dx = \int_{-3}^0 |x| \, dx + \int_0^b |x| \, dx.$

• On the interval $[-3, 0]$, $|x| = -x$, so the integral becomes:

$\int_{-3}^0 |x| \, dx = \int_{-3}^0 -x \, dx = \left[ -\frac{x^2}{2} \right]_{-3}^0 = \left( -\frac{0^2}{2} \right) - \left( -\frac{(-3)^2}{2} \right) = 0 - \left( -\frac{9}{2} \right) = \frac{9}{2}.$

• On the interval $[0, b]$, $|x| = x$, so the integral becomes:

$\int_0^b |x| \, dx = \int_0^b x \, dx = \left[ \frac{x^2}{2} \right]_0^b = \frac{b^2}{2}.$

Thus, the total integral is:

$\int_{-3}^b |x| \, dx = \frac{9}{2} + \frac{b^2}{2}.$

We are given that this integral equals 45:

$\frac{9}{2} + \frac{b^2}{2} = 45.$

Multiply through by 2 to eliminate the fractions:

$9 + b^2 = 90.$

Solve for $b^2$:

$b^2 = 90 - 9 = 81.$

Thus,

$b = \pm 9.$

Since we assumed $b \geq 0$, we conclude that $b = 9$.

Step 3: Case 2 — $b < 0$

For $b < 0$, the absolute value function $|x| = -x$ for all $x$ in the interval $[-3, b]$. Therefore, the integral becomes:

$\int_{-3}^b |x| \, dx = \int_{-3}^b -x \, dx = \left[ -\frac{x^2}{2} \right]_{-3}^b = -\frac{b^2}{2} + \frac{(-3)^2}{2}.$

Simplifying:

$\int_{-3}^b |x| \, dx = -\frac{b^2}{2} + \frac{9}{2} = \frac{9 - b^2}{2}.$

We are given that this integral equals 45:

$\frac{9 - b^2}{2} = 45.$

Multiply both sides by 2:

$9 - b^2 = 90.$

Solve for $b^2$:

$-b^2 = 90 - 9 = 81,$

which implies

$b^2 = -81.$

Since this is not possible for real numbers, there are no solutions when $b < 0$.

Final Answer:

The only solution is $b = 9$.

Additional Information:

Would you like further details or an explanation on any steps?

Here are some related questions you may find interesting:

1. How do the properties of the absolute value function affect integrals?
2. What would happen if we changed the bounds of the integral?
3. How do we handle integrals involving piecewise functions?
4. What happens if we have different powers of $|x|$ in the integral?
5. Can you explain how the method of splitting the integral works for other piecewise functions?

Tip: When dealing with absolute value functions in integrals, always consider splitting the integral based on where the function changes, which is usually at $x = 0$ for the absolute value.