Shaded regions pop up everywhere—textbook graphs, AP® free-response problems, and even window designs. Therefore, understanding how to find the area of the shaded region is a must-have skill. This lesson explains the process by using definite integrals, the official AP® objective “8.5 area between curves,” and a clear five-step strategy.

Why Shaded Regions Matter on the AP® Exam

On the AP® Calculus AB exam, an area-between-curves task often earns 4–6 points. BC students see it again inside volume, work, and probability prompts. Consequently, mastering shaded regions pays dividends.

Important words will appear repeatedly:

• region
• boundary curve
• slice (vertical or horizontal)
• integrand

Knowing them streamlines every solution.

The Fundamental Idea: Area as “Top – Bottom” or “Right – Left”

At its core, a shaded region is the sum of thin rectangles. As the rectangles shrink, Riemann sums become a definite integral.

Formula boxes:

• Vertical slices
  • A = \displaystyle\int_{a}^{b} \big(\text{top} - \text{bottom}\big)dx
• Horizontal slices
  • A = \displaystyle\int_{c}^{d} \big(\text{right} - \text{left}\big)dy

Decision checklist:

1. Sketch the region.
2. If one pair of curves is always “top” and “bottom,” choose vertical slices.
3. However, if the curves trade top-bottom roles, horizontal slices may avoid multiple integrals.

The Five-Step Strategy to Find the Area of the Shaded Region

1. Sketch or analyze the graph; clearly label the shaded region.
2. Choose a slicing direction that minimizes piece-wise work.
3. Find all intersection points; these become limits of integration.
4. Write the integrand as “top – bottom” or “right – left.”
5. Evaluate the definite integral, then state units if provided.

Find the Area of the Shaded Region Example 1: Vertical Slices (Classic “Top – Bottom”)

Problem: Find the area of the shaded region bounded by

y = 4 - x^{2} and y = x + 2.

Step-by-step solution

1. Graph notes
   • y = 4 - x^{2} is a downward parabola.
   • y = x + 2 is an increasing line.
   • The parabola stays above the line within the region.
2. Intersection points
   • Solve 4 - x^{2} = x + 2.
   • Rearrange: x^{2} + x - 2 = 0.
   • Factor: (x + 2)(x - 1) = 0.
   • Therefore, x = -2 or x = 1.
3. Limits of integration: a = -2, b = 1.
4. Integrand
   • Top curve – bottom curve → (4 - x^{2}) - (x + 2) = 2 - x - x^{2}.
5. Evaluate A = \displaystyle\int_{-2}^{1} (2 - x - x^{2})dx
   • Antiderivative: 2x - \dfrac{x^{2}}{2} - \dfrac{x^{3}}{3}
   • Substitute limits:
     • At x = 1: 2(1) - \dfrac{1}{2} - \dfrac{1}{3} = \dfrac{12 - 3 - 2}{6} = \dfrac{7}{6}
     • At x = -2: 2(-2) - \dfrac{4}{2} - \dfrac{-8}{3} = -4 - 2 + \dfrac{8}{3} = \dfrac{-18 + 8}{3} = \dfrac{-10}{3}
   • Difference: \dfrac{7}{6} - \big(\dfrac{-10}{3}\big) = \dfrac{7}{6} + \dfrac{20}{6} = \dfrac{27}{6} = \dfrac{9}{2}

Final answer: The shaded area equals \dfrac{9}{2}\text{square units}.

Find the Area of the Shaded Region Example 2: Horizontal Slices (Switch to y)

Problem

Find the area of the shaded region bounded by

x = y^{2} and x = 2y + 3.

Vertical slices would require two separate integrals because the curves cross. Therefore, horizontal slices are simpler.

Step-by-step solution

1. Intersection points (solve for y)
   • Set y^{2} = 2y + 3.
   • Rearrange: y^{2} - 2y - 3 = 0.
   • Factor: (y - 3)(y + 1) = 0.
   • Thus, y = 3 or y = -1.
2. Limits: c = -1, d = 3.
3. Right-minus-left integrated
   • Right curve: x = 2y + 3.
   • Left curve: x = y^{2}.
   • Difference: (2y + 3) - y^{2}.
4. Integral A = \displaystyle\int_{-1}^{3} \big((2y + 3) - y^{2}\big)dy
5. Evaluate
   • Antiderivative: y^{2} + 3y - \dfrac{y^{3}}{3}
     • At y = 3: 9 + 9 - \dfrac{27}{3} = 18 - 9 = 9
     • At y = -1: 1 - 3 - \dfrac{-1}{3} = -2 + \dfrac{1}{3} = -\dfrac{5}{3}
   • Difference: 9 - \big(-\dfrac{5}{3}\big) = \dfrac{27}{3} + \dfrac{5}{3} = \dfrac{32}{3}

Final answer: The shaded area equals \dfrac{32}{3}\text{square units}.

Find the Area of the Shaded Region Example 3: Piece-Wise Region (AP-Style)

Problem (verbal description)

A graph shows two curves crossing twice. Between x = 1 and x = 4 the region is shaded. From x = 1 to x = 2.5, the top curve is y = 6 - x. From x = 2.5 to x = 4, the top curve switches to y = \sqrt{x + 2}. The bottom curve throughout is y = x - 1.

Students must split the integral at x = 2.5:

A = \displaystyle\int_{1}^{2.5} \big[(6 - x) - (x - 1)\big]dx + \int_{2.5}^{4} \big[\sqrt{x + 2} - (x - 1)\big]dx

Step-by-step evaluation confirms an area of approximately 5.31 square units (calculator required). Remember to check calculator window settings to view both intersections.

Common Pitfalls and How to Avoid Them

• Curves may swap positions; therefore, always redraw “top – bottom” after an intersection.
• When switching from dx integrals to dy integrals, signs can flip; stay alert.
• Intersection points drive limits; mis-solving them shifts the entire answer.
• Modeling questions include units—square meters, square feet, etc.—so always attach them.

Quick Reference Chart: Key Terms & Definitions

Term	Definition	Symbol/Note
Shaded region	The exact slice of the plane whose area is requested	—
Boundary curve	Edge of the region; a function like y = f(x) or x = g(y)	f, g
Vertical slice	Thin rectangle Δx wide; height = top – bottom	Integrate dx
Horizontal slice	Thin rectangle Δy tall; width = right – left	Integrate dy
Intersection points	Points where curves meet; supply limits	Solve f = g
Definite integral	Limit of summed slice areas; yields exact area	\int

Conclusion

Finding the area of the shaded region relies on one powerful idea: slice the region, subtract the correct curves, and integrate. The five-step strategy—sketch, choose direction, locate intersections, write the integrand, and evaluate—works for every “8.5 area between curves” question. With regular practice, deciding between dx and dy becomes second nature, paving the way for future AP® topics like volumes of revolution and work problems.

Large or small, every shaded region can be conquered by applying these principles systematically. Therefore, keep the checklist handy, avoid the common pitfalls, and the next AP® graph will look far less intimidating.

Sharpen Your Skills for AP® Calculus AB-BC

Are you preparing for the AP® Calculus exam? We’ve got you covered! Try our review articles designed to help you confidently tackle real-world math problems. You’ll find everything you need to succeed, from quick tips to detailed strategies. Start exploring now!

• 8.4 Finding the Area Between Curves Expressed as Functions of x
• 8.6 Finding the Area Between Curves That Intersect at More Than Two Points

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