current attempt in progress given the figure and ∫−10f(x)d(x)=1.3, estimate : (a) ∫01f(x)dx= (b) ∫−11f(x)dx=…

Explanation:

Step1: Use integral property

We know that $\int_{a}^{c}f(x)dx=\int_{a}^{b}f(x)dx+\int_{b}^{c}f(x)dx$. So, $\int_{-1}^{1}f(x)dx=\int_{-1}^{0}f(x)dx+\int_{0}^{1}f(x)dx$. Also, the definite - integral $\int_{a}^{b}f(x)dx$ is the net signed area between the curve $y = f(x)$ and the $x$ - axis from $x=a$ to $x = b$, while the total shaded area is the sum of the absolute values of the positive and negative areas.

Step2: Estimate $\int_{0}^{1}f(x)dx$

From the graph, the area between $x = 0$ and $x = 1$ under the curve $y=f(x)$ is equal in magnitude but opposite in sign to the area between $x=-1$ and $x = 0$ (by symmetry or visual inspection). Since $\int_{-1}^{0}f(x)dx = 1.3$, then $\int_{0}^{1}f(x)dx=-1.3$.

Step3: Calculate $\int_{-1}^{1}f(x)dx$

Using the property $\int_{-1}^{1}f(x)dx=\int_{-1}^{0}f(x)dx+\int_{0}^{1}f(x)dx$, substitute $\int_{-1}^{0}f(x)dx = 1.3$ and $\int_{0}^{1}f(x)dx=-1.3$. So, $\int_{-1}^{1}f(x)dx=1.3+( - 1.3)=0$.

Step4: Calculate the total shaded area

The total shaded area is the sum of the absolute - value of the area above the $x$ - axis and the absolute - value of the area below the $x$ - axis. The area above the $x$ - axis (from $x=-1$ to $x = 0$) is $|\int_{-1}^{0}f(x)dx| = 1.3$, and the area below the $x$ - axis (from $x = 0$ to $x = 1$) is $|\int_{0}^{1}f(x)dx|=1.3$. So the total shaded area is $1.3 + 1.3=2.6$.

Answer:

(a) $-1.3$ (b) $0$ (c) $2.6$