Unit 6 Jeopardy:Integration and Accumulation of Change

Riemann Sums

100

Using a Left Riemann Sum, find the area under the curve over the given interval using 4 rectangles.

y=x²-2x+3 ; [0,4]

14

200

Using a Right Riemann Sum, find the area under the curve over the given interval using 4 rectangles.

y=-2x+10 ; [1,5]

12

300

Using a Midpoint Sum, find the area under the curve over the given interval using 4 rectangles.

y=x²+2 ; [-1,3]

23

400

Using a Trapezoidal Sum, find the area under the curve over the given interval using 4 rectangles.

y=-x²+6 ; [-1,3]

8

500

Using a Midpoint Sum, find the area under the curve over the given interval using 4 rectangles.

y=sin(x) ; [0,π]

2.053

Indefinite Integrals

100

Evaluate the Indefinite Integral

∫6x(x²+1)⁵dx

(x²+1)⁶+C

200

Evaluate the Indefinite Integral

∫3/(2x+1)dx

(3/2)ln(2x+1)+C

300

Evaluate the Indefinite Integral

∫(12x²)/(x³-5)dx

4ln(x³-5)+C

400

Evaluate the Indefinite Integral

∫(5x⁴)/(x⁵+2)dx

ln(x⁵+2)+C

500

Evaluate the Indefinite Integral

∫(4x³)/(√2x⁴+7)dx

(√2x⁴+7)+C

Find F^prime(x)

100

Find F^prime(x)

F(x)=∫(3t+2)dt ; [1,x]

3x+2

200

Find F^prime(x)

F(x)=∫(2/t²)dt ; [1,x]

2/x²

300

Find F^prime(x)

F(x)=∫(t²+1)dt ; [0,2x]

8x²+2

400

Find F^prime(x)

F(x)=∫(1/t+1)dt ; [2,3x]

3/(3x+1)

500

Find F^prime(x)

F(x)=∫(ln(t)+1)dt ; [0,√x]

(ln(√x)+1)(1/2√x)

Mean Value Theorem + Average Value

100

Find the average value of the function over the given interval.

f(x)=x+4 ; [0,6]

f_avg=7

200

Find the values of c that satisfy the Mean Value Theorem for Integrals

f(x)=x+1 ; [0,4]

c=2

300

Find the values of c that satisfy the Mean Value Theorem for Integrals

f(x)=e^x ; [0,1]

c=ln(e-1)

400

Find the average value of the function over the given interval.

f(x)=x√(x+1) ; [0,3]

f_avg=116/45

500

Find the values of c that satisfy the Mean Value Theorem for Integrals

f(x)=cos(x) ; [0,π]

c=π/2