Applications of Definite Integral — MHT-CET Maths

Formulas (5)

• Area Under a Curve Is an Integral of |y| · Area under a curve
  $$A = \int_a^b\left|f(x)\right|dx \qquad (f \ge 0 \text{ on } [a, b] \Rightarrow A = \int_a^b f(x)\,dx)$$
• When the Curve Crosses the Axis: Split and Take Each Piece Positive · Area across a sign change
  $$A = \left|\int_a^c f\,dx\right| + \left|\int_c^b f\,dx\right| \quad (f(c) = 0)$$
• Curves Given as x = g(y): Integrate in y · Horizontal strips
  $$A = \int_c^d\left|g(y)\right|dy \qquad \text{(strip width } x = g(y)\text{, thickness } dy)$$
• Recover the Curve First: Unknown Coefficients and a Given Derivative · Two conditions, two unknowns
  $$\text{point: } y_0 = f(x_0;\,a, b) \qquad \text{area: } \int_0^{4}f(x;\,a, b)\,dx = 8$$
• Dividing an Area in Half, and the Integral as Accumulated Change · Halving and accumulation
  $$\int_0^\alpha f\,dx = \frac12\int_0^b f\,dx \qquad P(b) - P(a) = \int_a^b\frac{dP}{dx}\,dx$$

Watch out for (5)

• Integrating over the wrong interval
  → Area Under a Curve Is an Integral of |y|
• Reporting the signed integral as the area
  → When the Curve Crosses the Axis: Split and Take Each Piece Positive
• Forgetting the lower half of a sideways parabola
  → Curves Given as x = g(y): Integrate in y
• Answering a when a − b was asked
  → Recover the Curve First: Unknown Coefficients and a Given Derivative
• Reporting the change instead of the new level
  → Dividing an Area in Half, and the Integral as Accumulated Change

Formulas (4)

• Solve for the Intersections, Then Integrate Upper Minus Lower · Area between two curves
  $$A = \int_{x_1}^{x_2}\left(y_{\text{upper}} - y_{\text{lower}}\right)dx, \qquad x_1, x_2 \text{ the roots of } y_1(x) = y_2(x)$$
• Two Parabolas Through the Origin, and Regions With Mirror Symmetry · Two standard parabola results
  $$\text{Area}\left(y^2 = 4ax,\ x^2 = 4ay\right) = \frac{16a^2}{3} \qquad \text{Area}\left(y = ax^2,\ x = ay^2\right) = \frac{1}{3a^2}$$
• Horizontal Strips: Integrate (Right − Left) in y · Horizontal strips between two curves
  $$A = \int_{y_1}^{y_2}\left(x_{\text{right}}(y) - x_{\text{left}}(y)\right)dy$$
• Regions Described by Several Inequalities: Sketch, Then Split Where the Top Changes · Piecewise ceiling
  $$A = \int_{x_0}^{x_1}\left(\text{ceiling}_1 - \text{floor}\right)dx + \int_{x_1}^{x_2}\left(\text{ceiling}_2 - \text{floor}\right)dx$$

Watch out for (4)

• Lower minus upper
  → Solve for the Intersections, Then Integrate Upper Minus Lower
• Counting a branch that is not there
  → Two Parabolas Through the Origin, and Regions With Mirror Symmetry
• Vertical strips on a sideways parabola
  → Horizontal Strips: Integrate (Right − Left) in y
• One ceiling for the whole interval
  → Regions Described by Several Inequalities: Sketch, Then Split Where the Top Changes

Formulas (4)

• The Circle Integral ∫√(a² − x²) dx, Quarter Discs and Sectors · Circle integral and sector
  $$\int\sqrt{a^2 - x^2}\,dx = \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2}\sin^{-1}\frac{x}{a} \qquad \text{sector} = \frac12 r^2\theta$$
• The Smaller Segment of a Circle Cut by a Vertical Line · Minor segment
  $$A = 2\int_c^a\sqrt{a^2 - x^2}\,dx = a^2\theta - \frac{a^2}{2}\sin 2\theta, \quad \cos\theta = \frac{c}{a}$$
• Ellipse Arc Minus Chord: Quarter-Ellipse Minus Triangle · Arc minus chord
  $$A = \frac{\pi ab}{4} - \frac{ab}{2} = \frac{ab}{4}(\pi - 2)$$
• Hyperbola Segments and Circle-Parabola Regions · Hyperbola integral and the circle-parabola region
  $$\int\sqrt{x^2 - a^2}\,dx = \frac{x}{2}\sqrt{x^2 - a^2} - \frac{a^2}{2}\log\left|x + \sqrt{x^2 - a^2}\right| \qquad \{x^2 + y^2 \le 1,\ y^2 \le 1 - x\}:\ \frac{\pi}{2} + \frac43$$

Watch out for (4)

• Reading x = y√3 as a 60° line
  → The Circle Integral ∫√(a² − x²) dx, Quarter Discs and Sectors
• Forgetting the factor 2 for the lower half
  → The Smaller Segment of a Circle Cut by a Vertical Line
• Using πab/2 for the quadrant
  → Ellipse Arc Minus Chord: Quarter-Ellipse Minus Triangle
• Setting up the circle-parabola region as one integral
  → Hyperbola Segments and Circle-Parabola Regions