Express as one or more integrals, BUT DO NOT EVALUATE, the area of the shaded region. Note: you do NOT need to determine the points of intersection. Just use the letters indicated on the graph. 3.0 2.5 y = 3sin(2x) (e,f) 2.0 y = tan-x 1.5 1.0 (a,b) 0.5 (c,d) y= 1-x2 0.2 0.4 0.6 0.8 1.0
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![**Integrating to Find the Area of the Shaded Region**
The task is to express the area of the shaded region as one or more integrals without evaluating them. The graph presents three functions and their interplay creates a bounded shaded region.
Functions involved:
1. \( y = 3 \sin(2x) \) - represented as an orange curve.
2. \( y = \tan^2 x \) - represented as a blue curve.
3. \( y = 1 - x^2 \) - represented as a green curve.
Points of interest on the graph are labeled as follows:
- Intersection at point \( (a, b) \)
- Intersection at point \( (c, d) \)
- Intersection at point \( (e, f) \)
**Graph Analysis and Integrals:**
The shaded region is enclosed by the function curves from the intervals determined by the intersection points. To express the area as integrals:
1. **From \( x = a \) to \( x = c \):** The area between \( y = \tan^2 x \) (upper boundary) and \( y = 3 \sin(2x) \) (lower boundary).
\[
\int_{a}^{c} \left( \tan^2 x - 3 \sin(2x) \right) \, dx
\]
2. **From \( x = c \) to \( x = e \):** The area between \( y = \tan^2 x \) (upper boundary) and \( y = 1 - x^2 \) (lower boundary).
\[
\int_{c}^{e} \left( \tan^2 x - (1 - x^2) \right) \, dx
\]
These integrals jointly represent the entire shaded region between the curves within the plotting limits. You do not need to calculate these integrals, as the problem specifies not to evaluate them. Instead, focus on expressing the boundaries and intersection points through integration.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1e69a358-0e29-48c6-949e-96b655159f87%2F5861349d-ea98-4654-9cab-28c18fb86c0b%2Fws45lq9_processed.jpeg&w=3840&q=75)
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