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Lesson Worksheet: Properties of Definite Integrals Mathematics • Higher Education

In this worksheet, we will practice using properties of definite integration, such as the order of integration limits, zero-width limits, sums, and differences.

Q1:

If ๏„ธ๐‘”(๐‘ฅ)๐‘ฅ=10๏Šฎ๏Šฑ๏Šญd, determine the value of ๏„ธ7๐‘”(๐‘ฅ)๐‘ฅ๏Šฑ๏Šญ๏Šฎd.

Q2:

The function ๐‘“ is continuous on [โˆ’4,4] and satisfies ๏„ธ๐‘“(๐‘ฅ)๐‘ฅ=9๏Šช๏Šฆd. Determine ๏„ธ[๐‘“(๐‘ฅ)โˆ’6]๐‘ฅ๏Šช๏Šฆd.

Q3:

If ๏„ธ๐‘“(๐‘ฅ)๐‘ฅ=82๏Šซ๏Šฑ๏Šชd and ๏„ธ๐‘”(๐‘ฅ)๐‘ฅ=74๏Šซ๏Šฑ๏Šชd, find ๏„ธ[2๐‘“(๐‘ฅ)โˆ’4๐‘”(๐‘ฅ)]๐‘ฅ.๏Šซ๏Šฑ๏Šชd

Q4:

If ๏„ธ๐‘“(๐‘ฅ)๐‘ฅ=โˆ’2.4๏Šจ๏Šฑ๏Šซd and ๏„ธ๐‘“(๐‘ฅ)๐‘ฅ=โˆ’1.4๏Šฑ๏Šง๏Šฑ๏Šซd, find ๏„ธ๐‘“(๐‘ฅ)๐‘ฅ๏Šจ๏Šฑ๏Šงd.

Q5:

The function ๐‘“ is continuous on โ„. Write ๏„ธ๐‘“(๐‘ฅ)๐‘ฅ+๏„ธ๐‘“(๐‘ฅ)๐‘ฅโˆ’๏„ธ๐‘“(๐‘ฅ)๐‘ฅ๏Šฉ๏Šฑ๏Šจ๏Šช๏Šฉ๏Šฆ๏Šฑ๏Šจddd in the form ๏„ธ๐‘“(๐‘ฅ)๐‘ฅ๏Œป๏Œบd.

  • A๏„ธ๐‘“(๐‘ฅ)๐‘ฅ๏Šช๏Šฆd
  • B๏„ธ๐‘“(๐‘ฅ)๐‘ฅ๏Šฆ๏Šชd
  • C๏„ธ๐‘“(๐‘ฅ)๐‘ฅ๏Šฉ๏Šฆd
  • D๏„ธ๐‘“(๐‘ฅ)๐‘ฅ๏Šฆ๏Šฉd
  • E๏„ธ๐‘“(๐‘ฅ)๐‘ฅ๏Šฑ๏Šจ๏Šฉd

Q6:

Suppose that on [โˆ’2,5], the values of ๐‘“ lie in the interval [๐‘š,๐‘€]. Between which bounds does ๏„ธ๐‘“(๐‘ฅ)๐‘ฅ๏Šซ๏Šฑ๏Šจd lie?

  • A7๐‘šโ‰ค๏„ธ๐‘“(๐‘ฅ)๐‘ฅโ‰ค7๏Šซ๏Šฑ๏Šจd
  • B7๐‘šโ‰ค๏„ธ๐‘“(๐‘ฅ)๐‘ฅโ‰ค7๐‘€๏Šซ๏Šฑ๏Šจd
  • C๐‘šโ‰ค๏„ธ๐‘“(๐‘ฅ)๐‘ฅโ‰ค๐‘€๏Šซ๏Šฑ๏Šจd
  • Dโˆ’2โ‰ค๏„ธ๐‘“(๐‘ฅ)๐‘ฅโ‰ค5๏Šซ๏Šฑ๏Šจd
  • E7โ‰ค๏„ธ๐‘“(๐‘ฅ)๐‘ฅโ‰ค7๐‘€๏Šซ๏Šฑ๏Šจd

Q7:

If the even function ๐‘“ is continuous over the interval [โˆ’4,4], where ๏„ธ๐‘“(๐‘ฅ)๐‘ฅ=2๏Šช๏Šฆd, determine the value of ๏„ธ๐‘“(๐‘ฅ)๐‘ฅ๏Šช๏Šฑ๏Šชd.

Q8:

The function ๐‘“ is odd, continuous on [โˆ’1,7], and satisfies ๏„ธ๐‘“(๐‘ฅ)๐‘ฅ=โˆ’17๏Šญ๏Šงd. Determine ๏„ธ๐‘“(๐‘ฅ)๐‘ฅ๏Šญ๏Šฑ๏Šงd.

Q9:

The function ๐‘“ is even, continuous on [โˆ’8,8], and satisfies ๏„ธ๐‘“(๐‘ฅ)๐‘ฅ=19๏Šฎ๏Šฑ๏Šฎd and ๏„ธ๐‘“(๐‘ฅ)๐‘ฅ=13๏Šช๏Šฆd. Determine ๏„ธ๐‘“(๐‘ฅ)๐‘ฅ๏Šฑ๏Šช๏Šฑ๏Šฎd.

  • A452
  • Bโˆ’32
  • Cโˆ’212
  • Dโˆ’72

Q10:

Determine ๏„ธ9๐‘ฅโˆ’6๐‘ฅ2๐‘ฅ+9๐‘ฅ๏Šง๏Šฑ๏Šง๏Šฉ๏Šจd.

This lesson includes 45 additional questions and 215 additional question variations for subscribers.

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