Math Problem Statement

Set up a sum of two integrals that equals the area of the shaded region bounded by the graphs of the functions f and g on​ [a,c] in the figure. Assume the curves intersect at xequals=b. a b c 0 x y y equals f left parenthesis x right parenthesisy=f(x)y equals g left parenthesis x right parenthesisy=g(x)

A curved line labeled y equals f of x starts at x equals A and decreases at a decreasing rate until it reaches the point where x equals B. The curve then increases at a decreasing rate until it reaches a maximum where x equals C. A curved line labeled y equals g of x starts at x equals A and decreases at a decreasing rate until it reaches a minimum. The line then increases intersecting the line labeled y equals f of x where x equals B. The line continues to increase reaching a maximum and then decreases until it reaches a point where x equals C. Question content area bottom Part 1 Choose the correct answer below. A.Integral from a to b left parenthesis f left parenthesis x right parenthesis plus g left parenthesis x right parenthesis right parenthesis dx minus Integral from b to c left parenthesis f left parenthesis x right parenthesis plus g left parenthesis x right parenthesis right parenthesis dx Integral from a to b left parenthesis f left parenthesis x right parenthesis plus g left parenthesis x right parenthesis right parenthesis dx minus Integral from b to c left parenthesis f left parenthesis x right parenthesis plus g left parenthesis x right parenthesis right parenthesis dx∫ab(f(x)+g(x))dx−∫bc(f(x)+g(x))dx B.Integral from a to b left parenthesis f left parenthesis x right parenthesis minus g left parenthesis x right parenthesis right parenthesis dx plus Integral from b to c left parenthesis g left parenthesis x right parenthesis minus f left parenthesis x right parenthesis right parenthesis dx Integral from a to b left parenthesis f left parenthesis x right parenthesis minus g left parenthesis x right parenthesis right parenthesis dx plus Integral from b to c left parenthesis g left parenthesis x right parenthesis minus f left parenthesis x right parenthesis right parenthesis dx∫ab(f(x)−g(x))dx+∫bc(g(x)−f(x))dx C.Integral from a to b left parenthesis g left parenthesis x right parenthesis minus f left parenthesis x right parenthesis right parenthesis dx plus Integral from b to c left parenthesis f left parenthesis x right parenthesis minus g left parenthesis x right parenthesis right parenthesis dx Integral from a to b left parenthesis g left parenthesis x right parenthesis minus f left parenthesis x right parenthesis right parenthesis dx plus Integral from b to c left parenthesis f left parenthesis x right parenthesis minus g left parenthesis x right parenthesis right parenthesis dx∫ab(g(x)−f(x))dx+∫bc(f(x)−g(x))dx D.Integral from b to c left parenthesis f left parenthesis x right parenthesis plus g left parenthesis x right parenthesis right parenthesis dx minus Integral from a to b left parenthesis f left parenthesis x right parenthesis plus g left parenthesis x right parenthesis right parenthesis dx Integral from b to c left parenthesis f left parenthesis x right parenthesis plus g left parenthesis x right parenthesis right parenthesis dx minus Integral from a to b left parenthesis f left parenthesis x right parenthesis plus g left parenthesis x right parenthesis right parenthesis dx∫bc(f(x)+g(x))dx−∫ab (f(x)+g(x))dx

Solution

The correct choice is B:

∫ab(f(x)−g(x))dx+∫bc(g(x)−f(x))dx\int_a^b \left( f(x) - g(x) \right) dx + \int_b^c \left( g(x) - f(x) \right) dx

Explanation:

  1. Understanding the bounds:

    • On the interval [a,b][a, b], f(x)f(x) lies above g(x)g(x), so the area is given by ∫ab(f(x)−g(x))dx\int_a^b \left( f(x) - g(x) \right) dx.
    • On the interval [b,c][b, c], g(x)g(x) lies above f(x)f(x), so the area is given by ∫bc(g(x)−f(x))dx\int_b^c \left( g(x) - f(x) \right) dx.

  2. Combining the areas:

    • To find the total area of the shaded region, sum the two integrals. Therefore, the total area is: ∫ab(f(x)−g(x))dx+∫bc(g(x)−f(x))dx\int_a^b \left( f(x) - g(x) \right) dx + \int_b^c \left( g(x) - f(x) \right) dx

This matches the choice B. Let me know if you would like further clarification or details!

Related Questions:

  1. What happens if f(x)=g(x)f(x) = g(x) at all points in [a,c][a, c]?
  2. How do you compute definite integrals if f(x)f(x) and g(x)g(x) are given explicitly?
  3. What does the result of each integral represent geometrically?
  4. How would this change if the intersection occurs at more than one point?
  5. Can the integral bounds or expressions change if f(x)f(x) and g(x)g(x) switch roles between intervals?

Tip:

Always ensure the function order in subtraction (f(x)−g(x)f(x) - g(x) or g(x)−f(x)g(x) - f(x)) matches the graph's visual hierarchy within the interval.

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Math Problem Analysis

Mathematical Concepts

Definite integrals
Area between curves
Graph analysis

Formulas

Area between two curves: ∫[a,b] (f(x) - g(x)) dx

Theorems

Fundamental Theorem of Calculus
Area interpretation of definite integrals

Suitable Grade Level

Grades 10-12

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