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Calculus, all content (2017 edition)

Course: Calculus, all content (2017 edition) > Unit 4

Lesson 6: Definite integral properties
Math
Calculus, all content (2017 edition)
Integration
Definite integral properties
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Warmup: Definite integral properties (no graph)

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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

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