Question

The area of the region bounded by $y=|x|$, $y=0$, $x=3$ and $x=-3$ is (in square units)

• A. 3
• B. 6
• C. 7
• D. 9
• E. 10

Hint:

Calculus Tip: Since $y = |x|$ is an even function, you can just evaluate $2 \int_0^3 x dx = 2 [\frac{x^2}{2}]_0^3 = [x^2]_0^3 = 9 - 0 = 9$.

Answer 1

The Correct Option is D

Concept:
The area bounded by a curve and the x-axis can be found using definite integrals, or by using basic geometry if the shapes are simple. The graph of $y = |x|$ forms two symmetric triangles with the x-axis ($y=0$) between $x = -3$ and $x = 3$.

Step 1: Visualize the geometric shape.
The function $y = |x|$ creates a V-shape starting at the origin $(0,0)$. The boundaries $x = -3$ and $x = 3$ cut off the V-shape, creating two right-angled triangles sitting on the x-axis.

Step 2: Find the dimensions of the first triangle (left side).
The left triangle goes from $x = -3$ to $x = 0$. Base = $3$ units. Height at $x = -3$ is $y = |-3| = 3$ units.

Step 3: Calculate the area of the left triangle.
Using the triangle area formula $A = \frac{1}{2}bh$: $$A_{left} = \frac{1}{2}(3)(3) = \frac{9}{2} = 4.5$$

Step 4: Calculate the area of the right triangle.
Because of the absolute value function's symmetry, the right triangle (from $x=0$ to $x=3$) is identical. $$A_{right} = 4.5$$

Step 5: Sum the areas.
Add the areas of the two triangles together to get the total bounded area: $$\text{Total Area} = A_{left} + A_{right} = 4.5 + 4.5 = 9$$ Hence the correct answer is (D) 9.