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.

  • A5360
  • B−140
  • C−760
  • D760
  • E−5360

Q3:

Evaluate 8𝑣+12𝑣+3𝑣d.

Q4:

Evaluate −5𝑒𝑥d.

  • A15𝑒
  • B−15𝑒
  • C30𝑒
  • D0
  • E−30𝑒

Q5:

Determine −6𝑥−3𝑥d.

Q6:

Determine 3𝑥−3𝑥+1𝑥d.

  • A486
  • B32
  • 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−92
  • B−6
  • C112
  • D92
  • E6

Q9:

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

Q10:

Determine (4+𝜋9𝑥)𝑥cosd.

  • A√2𝜋2+4𝜋3
  • B−14𝜋9−√2𝜋18
  • C−4𝜋3−√2𝜋2
  • D−4𝜋9−√2𝜋18
  • E√2𝜋18+4𝜋9

Q11:

Determine 6𝜋𝑧4𝑧sind.

  • A24√2𝜋
  • B−24√2𝜋
  • C4√2𝜋
  • D24√2

Q12:

Evaluate −3𝜃𝜃cscd.

  • A−3+3√3
  • B−3√22+3√32
  • C−√3
  • D−3√3+3
  • E−3√32+3√22

Q13:

Evaluate −4𝑥𝜃sind.

  • A6
  • B−4−2√3
  • C−6
  • D−2√3−3
  • E2√3+4

Q14:

Evaluate 2𝑡𝑡𝑡csccotd.

  • A−2
  • B0
  • C2
  • D−2√3
  • E2√3

Q15:

Determine 85𝜃𝜃cosd.

  • A20√2
  • B−4√25
  • C4√2
  • D4√25

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
  • B4373
  • C−4373
  • D54
  • E−54

Q18:

Determine 5𝑥+8𝑥7𝑥𝑥d.

  • A514𝑥+87𝑥+C
  • B57𝑥+87𝑥+C
  • C107𝑥+87𝑥+C
  • D87𝑥+514+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−962ln
  • B962ln
  • C192
  • D−192
  • E962log

Q21:

Evaluate 3𝑧𝑧d.

  • A−3√6+4√3
  • B8√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.

  • A3883
  • B342
  • C2263
  • D3563

Q24:

Determine 4𝑒𝑧d.

  • A−23𝑒+C
  • B4𝑒+C
  • C−23𝑒+C
  • D−23𝑒+C

Q25:

If 𝑓(𝑥)=6|𝑥|, determine 𝑓(𝑥)𝑥d.

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