(1 point)
Suppose the region on the left in the figure (with blue shading) has area is 33 , and the region on the right (with green shading) has area 3. Using the graph of $f(x)$ in the figure, find the following integrals.
\[
\begin{array}{l}
\int_{a}^{b} f(x) d x=\square \\
\int_{b}^{c} f(x) d x=\square \\
\int_{a}^{c} f(x) d x=\square \\
\int_{a}^{c}|f(x)| d x=\square
\end{array}
\]
Graph of $y=f(x)$
Answer
Final Answer: \[\begin{array}{l}\int_{a}^{b} f(x) d x=\boxed{33} \\int_{b}^{c} f(x) d x=\boxed{-3} \\int_{a}^{c} f(x) d x=\boxed{30} \\int_{a}^{c}|f(x)| d x=\boxed{36}\end{array}\]
Step 1 ：The integral of a function over an interval is the signed area under the curve of the function over that interval. The sign of the area is determined by whether the function is above or below the x-axis over the interval. If the function is above the x-axis, the area is positive. If the function is below the x-axis, the area is negative.
Step 2 ：The integral of the absolute value of a function over an interval is the total area under the curve of the function over that interval, regardless of whether the function is above or below the x-axis.
Step 3 ：The integral of \(f(x)\) from \(a\) to \(b\) is the area under the curve from \(a\) to \(b\), which is given as 33.
Step 4 ：The integral of \(f(x)\) from \(b\) to \(c\) is the area under the curve from \(b\) to \(c\), which is given as 3. However, since the function is below the x-axis over this interval, the area is negative. Therefore, the integral is -3.
Step 5 ：The integral of \(f(x)\) from \(a\) to \(c\) is the sum of the integrals from \(a\) to \(b\) and from \(b\) to \(c\), which is 33 + (-3) = 30.
Step 6 ：The integral of the absolute value of \(f(x)\) from \(a\) to \(c\) is the total area under the curve from \(a\) to \(c\), which is 33 + 3 = 36.
Step 7 ：Final Answer: \[\begin{array}{l}\int_{a}^{b} f(x) d x=\boxed{33} \\int_{b}^{c} f(x) d x=\boxed{-3} \\int_{a}^{c} f(x) d x=\boxed{30} \\int_{a}^{c}|f(x)| d x=\boxed{36}\end{array}\]
