### Related Formula textArea = int_a^b (y_textupper - y_textlower) \, dx$\text{Area} = \int_{a}^{b} (y_{\text{upper}} - y_{\text{lower}}) \, dx$ ### Core Logic Plotting the boundary lines and tracking intersection points yields a composite geometric region bounding a central area.

### Step 1: Set up the integral pieces The bounded region A$A$ can be conceptualized as a total bounding box/rectangle minus specific external integrals: A = 4 - 2 int_0^1 left[ (3 - x) - left( fracx^2 + 32 right) right] \, dx$A = 4 - 2 \int_{0}^{1} \left[ (3 - x) - \left( \frac{x^2 + 3}{2} \right) \right] \, dx$ ### Step 2: Evaluate the Integral A = 4 - 2 left[ 3x - fracx^22 - fracx^36 - frac32x right]_0^1$A = 4 - 2 \left[ 3x - \frac{x^2}{2} - \frac{x^3}{6} - \frac{3}{2}x \right]_{0}^{1}$ A = 4 - 2 left[ 3 - frac12 - frac16 - frac32 right] = 4 - 2 left[ frac56 right] = 4 - frac53 = frac73$A = 4 - 2 \left[ 3 - \frac{1}{2} - \frac{1}{6} - \frac{3}{2} \right] = 4 - 2 \left[ \frac{5}{6} \right] = 4 - \frac{5}{3} = \frac{7}{3}$ ### Step 3: Calculate 6A 6A = 6 times frac73 = 14$6A = 6 \times \frac{7}{3} = 14$ ### Pattern Recognition When dealing with multiple absolute functions (|x|$|x|$, |x-1|$|x-1|$), check for coordinate mirror symmetry across vertical axes to slash total required calculus computations in half. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Mathematics: Application of Integrals