Questions: Suppose the region on the left in the figure (with blue shading) has area is 21, and the region on the right (with green shading) has area 7. Using the graph of f(x) in the figure, find the following integrals. ∫ab f(x) dx= □ ∫bc f(x) dx= □ ∫ac f(x) dx= □ ∫ac f(x) dx= □

Suppose the region on the left in the figure (with blue shading) has area is 21, and the region on the right (with green shading) has area 7. Using the graph of f(x) in the figure, find the following integrals.
∫ab f(x) dx= □
∫bc f(x) dx= □
∫ac f(x) dx= □
∫ac f(x) dx= □

Transcript text: Suppose the region on the left in the figure (with blue shading) has area is 21, and the region on the right (with green shading) has area 7. Using the graph of $f(x)$ in the figure, find the following integrals. $\int_{a}^{b} f(x) d x=$ $\square$ $\int_{b}^{c} f(x) d x=$ $\square$ $\int_{a}^{c} f(x) d x=$ $\square$ $\int_{a}^{c}|f(x)| d x=$ $\square$

Solution

Solution Steps

Step 1: Calculate $\int_a^b f(x)dx$

The integral $\int_a^b f(x)dx$ represents the area between the curve $f(x)$ and the x-axis from $x=a$ to $x=b$. In this case, the area is below the x-axis, so the integral is negative. The area is given as 21, so $\int_a^b f(x)dx = -21$.

Step 2: Calculate $\int_b^c f(x)dx$

The integral $\int_b^c f(x)dx$ represents the area between the curve $f(x)$ and the x-axis from $x=b$ to $x=c$. In this case, the area is above the x-axis, so the integral is positive. The area is given as 7, so $\int_b^c f(x)dx = 7$.

Step 3: Calculate $\int_a^c f(x)dx$

The integral $\int_a^c f(x)dx$ can be calculated by summing the two integrals above:

$\int_a^c f(x)dx = \int_a^b f(x)dx + \int_b^c f(x)dx = -21 + 7 = -14$.