Area Between Curves Practice

Sample questions

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1. Sample 1difficulty 2/5

   Area between $y = f(x)$ (upper) and $y = g(x)$ (lower) on $[a, b]$ equals

   • A

     $\int_a^b f\,dx + \int_a^b g\,dx$

   • B

     $f(b) - g(a)$

   • C

     $\int_a^b (f - g)\,dx$

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   • D

     $f(a) - g(b)$

   Why

   Vertical-strip integration of (upper − lower).

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2. Sample 2difficulty 2/5

   Find the area between $y = x$ and $y = x^2$ on $[0, 1]$.

   • A

     $1$

   • B

     $\tfrac{1}{3}$

   • C

     $\tfrac{1}{2}$

   • D

     $\tfrac{1}{6}$

     check_circle

   Why

   $\int_0^1 (x - x^2)\,dx = [x^2/2 - x^3/3]_0^1 = 1/2 - 1/3 = 1/6$.

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3. Sample 3difficulty 2/5

   Area between $y = x$ and $y = -x$ on $[0, 2]$:

   • A

     $0$

   • B

     $8$

   • C

     $2$

   • D

     $4$

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   Why

   $\int_0^2 (x - (-x))\,dx = \int_0^2 2x\,dx = 4$.

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4. Sample 4difficulty 2/5

   Region between $y = e^x$ and $y = 1$ on $[0, 1]$:

   • A

     $e - 2$

     check_circle
   • B

     $e + 1$

   • C

     $1 - e$

   • D

     $e - 1$

   Why

   $\int_0^1 (e^x - 1)\,dx = (e - 1) - 1 = e - 2$.

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5. Sample 5difficulty 2/5

   The shaded area between the curve and the line equals

   • A

     $\int_a^b (g - f)\,dx$

   • B

     $\int_a^b (f - g)\,dx$

     check_circle
   • C

     $\int_a^b f\,dx$

   • D

     $\int_a^b g\,dx$

   Why

   Vertical strips: (upper) − (lower) integrated.

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