Definite Integral worksheet

1. Definite integral (approximate): 42

1. Read your function the usual way, powers first, then multiplying, then adding (angles in radians)

   $\text{f(x) = 4 × x^1 + 3 × x}$

2. We want the area between the curve and the x-axis. Area above the axis counts as plus, and area below counts as minus

   $\text{∫ from 2 to 4 of (4 × x^1 + 3 × x) dx}$

3. Estimate the area piece by piece (Gauss–Kronrod method). Each piece is checked at 15 points and again at 7, and the gap between them shows the error. The worst piece is cut in half until the error is tiny. That took 1 piece and 15 function values

   $\text{estimated error ≈}0$

4. Check the answer a second way, with Simpson's rule on 200 equal strips

   $\text{Simpson ≈}42$

5. Find the curve's average height: divide the area by the width of the interval

   $\frac{42}{4-2}$$=21$

6. So the definite integral, a close estimate, is

   $\text{∫ f(x) dx ≈}42$

2. Definite integral (approximate): 20

1. Read your function the usual way, powers first, then multiplying, then adding (angles in radians)

   $\text{f(x) = 3 × x^1 + 2 × x}$

2. We want the area between the curve and the x-axis. Area above the axis counts as plus, and area below counts as minus

   $\text{∫ from 1 to 3 of (3 × x^1 + 2 × x) dx}$

3. Estimate the area piece by piece (Gauss–Kronrod method). Each piece is checked at 15 points and again at 7, and the gap between them shows the error. The worst piece is cut in half until the error is tiny. That took 1 piece and 15 function values

   $\text{estimated error ≈}0.000000000000004$

4. Check the answer a second way, with Simpson's rule on 200 equal strips

   $\text{Simpson ≈}20$

5. Find the curve's average height: divide the area by the width of the interval

   $\frac{20}{3-1}$$=10$

6. So the definite integral, a close estimate, is

   $\text{∫ f(x) dx ≈}20$

3. Definite integral (approximate): 20.67

1. Read your function the usual way, powers first, then multiplying, then adding (angles in radians)

   $\text{f(x) = 1 × x^2 + 3 × x}$

2. We want the area between the curve and the x-axis. Area above the axis counts as plus, and area below counts as minus

   $\text{∫ from 1 to 3 of (1 × x^2 + 3 × x) dx}$

3. Estimate the area piece by piece (Gauss–Kronrod method). Each piece is checked at 15 points and again at 7, and the gap between them shows the error. The worst piece is cut in half until the error is tiny. That took 1 piece and 15 function values

   $\text{estimated error ≈}0$

4. Check the answer a second way, with Simpson's rule on 200 equal strips

   $\text{Simpson ≈}20.6666666667$

5. Find the curve's average height: divide the area by the width of the interval

   $\frac{20.6666666667}{3-1}$$=10.3333333333$

6. So the definite integral, a close estimate, is

   $\text{∫ f(x) dx ≈}20.6666666667$

4. Definite integral (approximate): 24.6

1. Read your function the usual way, powers first, then multiplying, then adding (angles in radians)

   $\text{f(x) = 3 × x^4 + 4 × x}$

2. We want the area between the curve and the x-axis. Area above the axis counts as plus, and area below counts as minus

   $\text{∫ from 1 to 2 of (3 × x^4 + 4 × x) dx}$

3. Estimate the area piece by piece (Gauss–Kronrod method). Each piece is checked at 15 points and again at 7, and the gap between them shows the error. The worst piece is cut in half until the error is tiny. That took 1 piece and 15 function values

   $\text{estimated error ≈}0.000000000000004$

4. Check the answer a second way, with Simpson's rule on 200 equal strips

   $\text{Simpson ≈}24.6000000002$

5. Find the curve's average height: divide the area by the width of the interval

   $\frac{24.6}{2-1}$$=24.6$

6. So the definite integral, a close estimate, is

   $\text{∫ f(x) dx ≈}24.6$

5. Definite integral (approximate): 45

1. Read your function the usual way, powers first, then multiplying, then adding (angles in radians)

   $\text{f(x) = 4 × x^1 + 2 × x}$

2. We want the area between the curve and the x-axis. Area above the axis counts as plus, and area below counts as minus

   $\text{∫ from 1 to 4 of (4 × x^1 + 2 × x) dx}$

3. Estimate the area piece by piece (Gauss–Kronrod method). Each piece is checked at 15 points and again at 7, and the gap between them shows the error. The worst piece is cut in half until the error is tiny. That took 1 piece and 15 function values

   $\text{estimated error ≈}0.000000000000011$

4. Check the answer a second way, with Simpson's rule on 200 equal strips

   $\text{Simpson ≈}45$

5. Find the curve's average height: divide the area by the width of the interval

   $\frac{45}{4-1}$$=15$

6. So the definite integral, a close estimate, is

   $\text{∫ f(x) dx ≈}45$

6. Definite integral (approximate): 221

1. Read your function the usual way, powers first, then multiplying, then adding (angles in radians)

   $\text{f(x) = 5 × x^4 + 4 × x}$

2. We want the area between the curve and the x-axis. Area above the axis counts as plus, and area below counts as minus

   $\text{∫ from 2 to 3 of (5 × x^4 + 4 × x) dx}$

3. Estimate the area piece by piece (Gauss–Kronrod method). Each piece is checked at 15 points and again at 7, and the gap between them shows the error. The worst piece is cut in half until the error is tiny. That took 1 piece and 15 function values

   $\text{estimated error ≈}0.000000000000085$

4. Check the answer a second way, with Simpson's rule on 200 equal strips

   $\text{Simpson ≈}221.0000000004$

5. Find the curve's average height: divide the area by the width of the interval

   $\frac{221}{3-2}$$=221$

6. So the definite integral, a close estimate, is

   $\text{∫ f(x) dx ≈}221$

7. Definite integral (approximate): 136

1. Read your function the usual way, powers first, then multiplying, then adding (angles in radians)

   $\text{f(x) = 6 × x^3 + 4 × x}$

2. We want the area between the curve and the x-axis. Area above the axis counts as plus, and area below counts as minus

   $\text{∫ from 1 to 3 of (6 × x^3 + 4 × x) dx}$

3. Estimate the area piece by piece (Gauss–Kronrod method). Each piece is checked at 15 points and again at 7, and the gap between them shows the error. The worst piece is cut in half until the error is tiny. That took 1 piece and 15 function values

   $\text{estimated error ≈}0.000000000000142$

4. Check the answer a second way, with Simpson's rule on 200 equal strips

   $\text{Simpson ≈}136$

5. Find the curve's average height: divide the area by the width of the interval

   $\frac{136}{3-1}$$=68$

6. So the definite integral, a close estimate, is

   $\text{∫ f(x) dx ≈}136$

8. Definite integral (approximate): 36

1. Read your function the usual way, powers first, then multiplying, then adding (angles in radians)

   $\text{f(x) = 1 × x^2 + 2 × x}$

2. We want the area between the curve and the x-axis. Area above the axis counts as plus, and area below counts as minus

   $\text{∫ from 1 to 4 of (1 × x^2 + 2 × x) dx}$

3. Estimate the area piece by piece (Gauss–Kronrod method). Each piece is checked at 15 points and again at 7, and the gap between them shows the error. The worst piece is cut in half until the error is tiny. That took 1 piece and 15 function values

   $\text{estimated error ≈}0.000000000000011$

4. Check the answer a second way, with Simpson's rule on 200 equal strips

   $\text{Simpson ≈}36$

5. Find the curve's average height: divide the area by the width of the interval

   $\frac{36}{4-1}$$=12$

6. So the definite integral, a close estimate, is

   $\text{∫ f(x) dx ≈}36$

9. Definite integral (approximate): 15.4

1. Read your function the usual way, powers first, then multiplying, then adding (angles in radians)

   $\text{f(x) = 2 × x^4 + 2 × x}$

2. We want the area between the curve and the x-axis. Area above the axis counts as plus, and area below counts as minus

   $\text{∫ from 1 to 2 of (2 × x^4 + 2 × x) dx}$

3. Estimate the area piece by piece (Gauss–Kronrod method). Each piece is checked at 15 points and again at 7, and the gap between them shows the error. The worst piece is cut in half until the error is tiny. That took 1 piece and 15 function values

   $\text{estimated error ≈}0.000000000000004$

4. Check the answer a second way, with Simpson's rule on 200 equal strips

   $\text{Simpson ≈}15.4000000002$

5. Find the curve's average height: divide the area by the width of the interval

   $\frac{15.4}{2-1}$$=15.4$

6. So the definite integral, a close estimate, is

   $\text{∫ f(x) dx ≈}15.4$

10. Definite integral (approximate): 326.25

1. Read your function the usual way, powers first, then multiplying, then adding (angles in radians)

   $\text{f(x) = 5 × x^3 + 1 × x}$

2. We want the area between the curve and the x-axis. Area above the axis counts as plus, and area below counts as minus

   $\text{∫ from 1 to 4 of (5 × x^3 + 1 × x) dx}$

3. Estimate the area piece by piece (Gauss–Kronrod method). Each piece is checked at 15 points and again at 7, and the gap between them shows the error. The worst piece is cut in half until the error is tiny. That took 1 piece and 15 function values

   $\text{estimated error ≈}0.000000000000128$

4. Check the answer a second way, with Simpson's rule on 200 equal strips

   $\text{Simpson ≈}326.25$

5. Find the curve's average height: divide the area by the width of the interval

   $\frac{326.25}{4-1}$$=108.75$

6. So the definite integral, a close estimate, is

   $\text{∫ f(x) dx ≈}326.25$