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Warmup: Definite integral properties (no graph)

Apply the properties of definite integrals to evaluate definite integrals.

Problem 1

Given integral, start subscript, minus, 5, end subscript, start superscript, minus, 1, end superscript, f, left parenthesis, x, right parenthesis, d, x, equals, 4, integral, start subscript, 3, end subscript, start superscript, 5, end superscript, f, left parenthesis, x, right parenthesis, d, x, equals, 9, and integral, start subscript, minus, 5, end subscript, start superscript, 5, end superscript, f, left parenthesis, x, right parenthesis, d, x, equals, 10, find the following:
a) integral, start subscript, minus, 1, end subscript, cubed, f, left parenthesis, x, right parenthesis, d, x, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

b) integral, start subscript, minus, 1, end subscript, start superscript, 5, end superscript, f, left parenthesis, x, right parenthesis, d, x, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

c) integral, start subscript, minus, 5, end subscript, start superscript, minus, 1, end superscript, 7, f, left parenthesis, x, right parenthesis, d, x, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

d) integral, start subscript, 5, end subscript, start superscript, minus, 5, end superscript, f, left parenthesis, x, right parenthesis, d, x, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Challenge

e*) integral, start subscript, 3, end subscript, start superscript, 5, end superscript, left parenthesis, f, left parenthesis, x, right parenthesis, plus, 4, right parenthesis, d, x, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Problem 2

Given integral, start subscript, minus, 8, end subscript, start superscript, minus, 2, end superscript, g, left parenthesis, x, right parenthesis, d, x, equals, 11, integral, start subscript, minus, 8, end subscript, start superscript, 0, end superscript, g, left parenthesis, x, right parenthesis, d, x, equals, 5, and integral, start subscript, minus, 2, end subscript, start superscript, 8, end superscript, g, left parenthesis, x, right parenthesis, d, x, equals, minus, 5, find the following:
a) integral, start subscript, minus, 2, end subscript, start superscript, 0, end superscript, g, left parenthesis, x, right parenthesis, d, x, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

b) integral, start subscript, 0, end subscript, start superscript, 8, end superscript, g, left parenthesis, x, right parenthesis, d, x, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

c) integral, start subscript, 8, end subscript, start superscript, minus, 8, end superscript, 10, g, left parenthesis, x, right parenthesis, d, x, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Challenge

d*) integral, start subscript, 0, end subscript, start superscript, 8, end superscript, left parenthesis, g, left parenthesis, x, right parenthesis, plus, x, right parenthesis, d, x, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Want to join the conversation?

  • hopper cool style avatar for user KeilahMartin
    In the final challenge question, how do we know to use the graph of y = x?
    (3 votes)
    Default Khan Academy avatar avatar for user
    • leafers sapling style avatar for user Aukele
      The way I think about it is that a definite integral is asking for the area under the curve/graph of the function within the integral. For example, in most of the problems above, we're looking for the integral (area under the curve) of the function y=g(x). But when we need to split the integral into two in the last problem, we're left with the integral (area under the curve) of y=g(x) and the integral (area under the curve) of y=x, because x was on its own and can be considered the function by itself.
      (3 votes)
  • leaf green style avatar for user Haj Maati  Douidy
    Is there any way you could do this one for me? I got stuck
    Integral of e^2x * (e(x)+1)^1/2
    Thank you
    (0 votes)
    Default Khan Academy avatar avatar for user