Chapter 7 Problems

Thirty chapter problems: 10 from the original book and 20 added for more practice.

Read the chapter problems → · Chapter solutions →

1. Original problem

   Estimate the area under $f(x)=x^2$ from x = 0 to x = 2 using the left corner point method of approximation and n = 2 rectangles.

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2. Original problem

   Estimate the area under $f(x)=\sqrt{x}$ from x = 1 to x = 4 using the right corner point method and 3 rectangles.

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3. Original problem

   Estimate the area under the curve of $y=\cos (x)+2$ using the midpoint method from x = 0 to x = 2π, and 4 rectangles.

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4. Original problem

   Estimate the area under the curve of $y=x^4$ from x = -2 to x = 2 using the method of trapezoids and 2 trapezoids for estimation.

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5. Original problem

   Estimate the area between the curves $y=\sqrt{x}$ and $y=x^2$ on the interval from x = 0 to x = 1 using the left corner point approximation method and 4 rectangles.

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6. Original problem

   Estimate the area between the curves $y=e^{x-1}-1$ and $y=\ln (x)$ from x = 1 to x = 3 using the method of trapezoids and 4 polygons for approximation.

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7. Original problem · clarified

   Estimate the area underneath $y=-x^2+2$ from x = 0 to x = 2 using right corner point approximation and 8 rectangles.

   Clarification: Estimate geometric area between the curve and the x-axis, using the absolute value of each right-endpoint height.

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8. Original problem

   Estimate the area between $y=-x+2$ and $y=x^2$ from x = 0 to x = 1 using the midpoint method of approximation and 4 rectangles.

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9. Original problem

   Estimate the area under the curve $y=2^{-x}$ from x = 0 to x = ∞ using the right corner point method and with the base of each rectangle being equal to 1.

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10. Original problem · clarified

    Estimate the area under the curve $y=\frac{1}{x^2}$ from x = 0 to x = ∞ using the right corner point method and with the base of each rectangle being equal to 1.

    Clarification: Estimate the unit-width right-rectangle sum, whose sample points are 1,2,3,… . This is a sampled estimate; the exact improper area starting at x=0 diverges.

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11. Added problem

    Estimate the area under $f\left(x\right)=x$ on $[1,4]$ using three left rectangles.

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12. Added problem

    For $f\left(x\right)=2^{-x}$ on [0,3], use three right rectangles, compute the matching left sum, and give area bounds.

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13. Added problem

    Use two midpoint rectangles to estimate the area under $f\left(x\right)=x^2+1$ on [−1,1].

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14. Added problem

    Use three trapezoids for $f\left(x\right)=2x+3$ on [−1,2]. Explain why the result is exact.

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15. Added problem

    For a decreasing nonnegative function, explain which equal-width rectangle estimate is larger than the true area: left or right?

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16. Added problem

    At x=0,1,2,3 a table gives f=2,4,8,10. Assume f is nonnegative and increasing. Find left/right sums and area bounds on [0,3].

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17. Added problem

    A table gives (x,f(x))=(0,2),(1,4),(3,8). Estimate its area with a trapezoid on each interval.

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18. Added problem

    Use two midpoint rectangles for the area between $y=3$ and $y=x^2$ on [0,1].

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19. Added problem

    A function is nonnegative and concave up on an interval. Explain whether its trapezoidal estimate lies above or below the exact area.

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20. Added problem

    Use two midpoint rectangles for $f\left(x\right)=4-x^2$ on [0,2]. Explain the error direction using concavity.

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21. Added problem

    At x=0,1,2,3 a table gives f=0,−2,1,1. On [0,3], find the signed right sum and the right estimate of geometric area.

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22. Added problem

    Compare left sums with n=2 and n=4 for $f\left(x\right)=x^3$ on [0,2]. Why does the second estimate increase?

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23. Added problem

    For $f\left(x\right)=x$ on [0,1], write Rₙ using ${{\Sigma }_{k=1}}^{n}k=n(n+1)/2$ and find its limit.

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24. Added problem

    Compute left, right, midpoint, and trapezoidal estimates with n=2 for $f\left(x\right)=1/(1+x)$ on [0,2].

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25. Added problem

    Find all antiderivatives of $6x^2-4$.

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26. Added problem

    Find an antiderivative of $\sin \left(3x\right)$ and check it.

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27. Added problem

    Find F if $F\prime \left(x\right)=2x$ and $F\left(1\right)=6$.

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28. Added problem

    A student proposes $(1/2)\sin \left(2x\right)$ as an antiderivative of $2\cos \left(2x\right)$. Check and correct it.

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29. Added problem

    A velocity table gives 2,6,14 m/s at t=0,1,2 s. Use trapezoids to estimate the displacement over the two seconds.

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30. Added problem

    For the infinite unit-width right rectangle sum of $2^{-x}$ from zero, how many rectangles make its remaining tail less than 0.01?

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