Suppose the region on the left in the figure (with blue shading) has area is 15 , and the region on the right (with green shading) has area 5 . Using the graph of $f(x)$ in the figure, find the following integrals. Graph of $y=f(x)$ \[ \begin{array}{l} \int_{a}^{b} f(x) d x= \\ \int_{b}^{c} f(x) d x= \\ \int_{a}^{c} f(x) d x= \\ \int_{a}^{c}|f(x)| d x= \end{array} \] Note: You can earn partial credit on this problem.
Solution
Step 1 ：The integral of a function from a to b is the area under the curve of the function from a to b. 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 absolute value of the integral from a to c is the total area under the curve from a to c, regardless of whether the function is above or below the x-axis.
Step 3 ：The first integral is from a to b, which corresponds to the blue shaded area in the figure. The second integral is from b to c, which corresponds to the green shaded area in the figure. The third integral is from a to c, which is the sum of the first two integrals. The fourth integral is the absolute value of the integral from a to c, which is the total area under the curve from a to c.
Step 4 ：Since we know the areas of the blue and green shaded regions, we can use these values to find the values of the integrals. The integral from a to b is 15, the integral from b to c is -5 (since the function is below the x-axis in this region), the integral from a to c is 15 - 5 = 10, and the absolute value of the integral from a to c is |10| = 10.
Step 5 ：Final Answer: \n\[\begin{array}{l}\int_{a}^{b} f(x) d x= \boxed{15} \\int_{b}^{c} f(x) d x= \boxed{-5} \\int_{a}^{c} f(x) d x= \boxed{10} \\int_{a}^{c}|f(x)| d x= \boxed{20}\end{array}\]
