Area Under a Curve: Step-by-Step Guide
The area under a curve y = f(x) between x = a and x = b is given by the definite integral of f(x) from a to b, provided f(x) >= 0 on that interval. When f(x) dips below the x-axis, the integral counts that region as negative, so for a true geometric area you must split the interval at each zero of f and integrate the absolute value.
Math Racer skill guide · Calculus 2 · AP Calc BC / College
How to solve it, step by step
- Identify the function and the interval. Determine the integrand f(x) and the bounds a and b. Sketch the curve to see where it is above or below the x-axis on [a, b].\[A = \int_{a}^{b} f(x)\,dx\]
- Find any x-intercepts inside the interval. Solve f(x) = 0 for x in (a, b). These zeros split the region into pieces that lie entirely above or entirely below the x-axis.\[f(x) = 0 \;\Rightarrow\; x = c_1, c_2, \dots\]
- Find an antiderivative F(x). Compute F(x) such that F'(x) = f(x). Use power, exponential, trig, or substitution rules as needed.\[F(x) = \int f(x)\,dx\]
- Apply the Fundamental Theorem of Calculus. Evaluate F(b) - F(a) on each subinterval. For a true geometric area, take the absolute value on subintervals where f(x) < 0 and add.\[A = \sum_i \left| F(c_{i+1}) - F(c_i) \right|\]
- Verify with units and a sanity check. Confirm the answer is positive and roughly matches the size of a bounding rectangle (height x width) on the interval.\[0 < A \lesssim (b-a)\,\max_{[a,b]} f\]
Worked example
Find the area under f(x) = 4 - x^2 from x = -2 to x = 2.
Step 1: The function f(x) = 4 - x^2 is non-negative on [-2, 2] since 4 - x^2 >= 0 when |x| <= 2, so the integral equals the geometric area.
Step 2: Set up the definite integral A = integral from -2 to 2 of (4 - x^2) dx.
Step 3: Find an antiderivative: F(x) = 4x - x^3/3.
Step 4: Evaluate F(2) - F(-2) = (8 - 8/3) - (-8 + 8/3) = 16 - 16/3 = 48/3 - 16/3 = 32/3.
Step 5: The area is 32/3 square units, approximately 10.67.
Verification: A bounding rectangle of width 4 and height 4 has area 16, and the parabolic cap should be about two-thirds of that: (2/3)(16) = 32/3. Matches.
Tips and common mistakes
- Always sketch the curve first; signed area can mislead you if the curve crosses the x-axis.
- Split the integral at each zero of f(x) inside [a, b] when computing geometric area.
- Remember that integral from a to b of f(x) dx gives signed area; |F(b) - F(a)| only equals geometric area if f does not change sign.
- For symmetric integrands on symmetric intervals, exploit symmetry: integral from -a to a of an even function equals 2 times integral from 0 to a.
- Check units: dx has units of x, so the integral inherits units of (units of f) times (units of x).
- If you cannot find an elementary antiderivative, switch to numerical methods such as Simpson's rule or the trapezoid rule.
- Compare to a quick rectangle estimate (b - a) times f(midpoint) to catch sign or arithmetic errors.