Worksheet: The Fundamental Theorem of Calculus: Evaluating Definite Integrals

In this worksheet, we will practice using the fundamental theorem of calculus to evaluate definite integrals.

Q1:

Evaluate 4𝑥𝑥d.

Q2:

Evaluate 45𝑡+34𝑡−23𝑡𝑡d.

  • A 5 3 6 0
  • B − 1 4 0
  • C − 7 6 0
  • D 7 6 0
  • E − 5 3 6 0

Q3:

Evaluate 8𝑣+12𝑣+3𝑣d.

Q4:

Evaluate −5𝑒𝑥d.

  • A 1 5 𝑒
  • B − 1 5 𝑒
  • C 3 0 𝑒
  • D0
  • E − 3 0 𝑒

Q5:

Determine −6𝑥−3𝑥d.

Q6:

Determine 3𝑥−3𝑥+1𝑥d.

  • A486
  • B 3 2
  • C54
  • D − 9

Q7:

Let 𝑓(𝑥)=6𝑥+1. Evaluate the definite integral of 𝑓 from 𝑥=2 to 𝑥=3.

Q8:

Evaluate (𝑢+2)(𝑢+1)𝑢d.

  • A − 9 2
  • B − 6
  • C 1 1 2
  • D 9 2
  • E6

Q9:

Let 𝑓(𝑥)=4𝑥<1,4𝑥𝑥≥1.ifif Determine 𝑓(𝑥)𝑥d.

Q10:

Determine (4+𝜋9𝑥)𝑥cosd.

  • A √ 2 𝜋 2 + 4 𝜋 3
  • B − 1 4 𝜋 9 − √ 2 𝜋 1 8
  • C − 4 𝜋 3 − √ 2 𝜋 2
  • D − 4 𝜋 9 − √ 2 𝜋 1 8
  • E √ 2 𝜋 1 8 + 4 𝜋 9

Q11:

Determine 6𝜋𝑧4𝑧sind.

  • A 2 4 √ 2 𝜋
  • B − 2 4 √ 2 𝜋
  • C 4 √ 2 𝜋
  • D 2 4 √ 2

Q12:

Evaluate −3𝜃𝜃cscd.

  • A − 3 + 3 √ 3
  • B − 3 √ 2 2 + 3 √ 3 2
  • C − √ 3
  • D − 3 √ 3 + 3
  • E − 3 √ 3 2 + 3 √ 2 2

Q13:

Evaluate −4𝑥𝜃sind.

  • A6
  • B − 4 − 2 √ 3
  • C − 6
  • D − 2 √ 3 − 3
  • E 2 √ 3 + 4

Q14:

Evaluate 2𝑡𝑡𝑡csccotd.

  • A − 2
  • B0
  • C2
  • D − 2 √ 3
  • E 2 √ 3

Q15:

Determine 85𝜃𝜃cosd.

  • A 2 0 √ 2
  • B − 4 √ 2 5
  • C 4 √ 2
  • D 4 √ 2 5

Q16:

If 𝑤(𝑡) is the rate of growth of a child in grams per week, what does 𝑤(𝑡)𝑡d represent?

  • AIt represents the child’s weight (in grams) at the age of 3 weeks.
  • BIt represents the child’s weight (in grams) at the age of 5 weeks.
  • CIt represents the child’s weight (in grams) at the age of 1 week.
  • DIt represents the increase in the child’s weight (in grams) during the last week.
  • EIt represents the net change in the child’s weight (in grams) between the ages of 1 and 5 weeks.

Q17:

Evaluate 4−5𝑥√𝑥𝑥d.

  • A − 6
  • B 4 3 7 3
  • C − 4 3 7 3
  • D54
  • E − 5 4

Q18:

Determine 5𝑥+8𝑥7𝑥𝑥d.

  • A 5 1 4 𝑥 + 8 7 𝑥 +  C
  • B 5 7 𝑥 + 8 7 𝑥 +  C
  • C 1 0 7 𝑥 + 8 7 𝑥 +  C
  • D 8 7 𝑥 + 5 1 4 +   C

Q19:

If oil leaks from a tank at a rate of 𝑟(𝑡) gallons per minute at time 𝑡, which of the following does 𝑟(𝑡)𝑡d represent?

  • AThe number of gallons of oil left in the tank after 120 minutes
  • BThe number of gallons of oil leaked between the 20th and 120th minutes
  • CThe number of gallons of oil left in the tank after 20 minutes
  • DThe number of gallons of oil left in the tank after 70 minutes
  • EThe number of gallons of oil leaked during the last minute

Q20:

Evaluate 4𝑠d.

  • A − 9 6 2 l n
  • B 9 6 2 l n
  • C192
  • D − 1 9 2
  • E 9 6 2 l o g

Q21:

Evaluate 3𝑧𝑧d.

  • A − 3 √ 6 + 4 √ 3
  • B 8 √ 3 + 6 √ 6
  • C − 8 √ 3 + 6 √ 6
  • D − 6 √ 6 + 8 √ 3
  • E − 4 √ 3 + 3 √ 6

Q22:

A honeybee population starts with 100 bees and increases at a rate of 𝑚(𝑡) bees per week. What does 100+𝑚(𝑡)𝑡d represent?

  • AIt represents the total population in week 16.
  • BIt represents the net change in population between week 1 and week 16.
  • CIt represents the net change in population between week 1 and week 8.
  • DIt represents the total population in week 1.
  • EIt represents the total population in week 8.

Q23:

Evaluate ||𝑥−4||𝑥d.

  • A 3 8 8 3
  • B342
  • C 2 2 6 3
  • D 3 5 6 3

Q24:

Determine 4𝑒𝑧d.

  • A − 2 3 𝑒 +      C
  • B 4 𝑒 +    C
  • C − 2 3 𝑒 +    C
  • D − 2 3 𝑒 +      C

Q25:

If 𝑓(𝑥)=6|𝑥|, determine 𝑓(𝑥)𝑥d.

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