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📝 Area problem and limits (12 MCQs)

📖 From Calculus • 2. Limits and Continuity an Introduction • 12 questions available

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What is Area problem and limits?

Definition:
The area problem seeks the exact area under a curve over a closed interval by summing the areas of rectangles and taking the limit as the number of rectangles approaches infinity. This process is formalized as $A = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x$ , where $\Delta x = \frac{b-a}{n}$ and $x_i^*$ is a sample point in each subinterval. It transforms approximate sums into precise integrals.

Example:
Find the area under $f(x) = x$ from $x=0$ to $x=2$ .
Solution: $A = \lim_{n \to \infty} \sum_{i=1}^n \left(\frac{2i}{n}\right)\left(\frac{2}{n}\right) = \lim_{n \to \infty} \frac{4}{n^2} \cdot \frac{n(n+1)}{2} = \lim_{n \to \infty} 2(1+\frac{1}{n}) = 2$ .

Reason:
This problem establishes the definite integral, linking summation and limits, which is crucial for computing total accumulation, such as distance from velocity or work from force, in real-world applications.

Easy

Medium

Hard

📝 All Area problem and limits MCQs

Q1. If the area under $f(x)=x^{2}$ from $x=0$ to $x=2$ is $\frac{8}{3}$ , what is the area from $x=0$ to $x=4$ ?

\frac{8}{3}

\frac{16}{3}

\frac{32}{3}

\frac{64}{3}

✅

💡 Difficulty: easy | ✅ Correct: D

📖 Explanation: The antiderivative of $x^{2}$ is $\frac{1}{3}x^{3}$ . Evaluating from 0 to 4 gives $\frac{1}{3}(4)^{3}= \frac{64}{3}$ . Because the integral scales with the cube of the endpoint, the area grows from $\frac{8}{3}$ to $\frac{64}{3}$ , confirming choice D.

Q2. Compare the total area between $y=\sin x$ on $[0,\pi]$ and the x‑axis with the area between $y=\cos x$ on $[0,\frac{\pi}{2}]$ . Which is larger?

A.Area under

\sin x

is larger ✅

B.Area under

\cos x

is larger

C.Both areas are equal

D.Cannot be determined without computation

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: $\int_{0}^{\pi}\sin x\,dx =[-\cos x]_{0}^{\pi}=2$ . Meanwhile $\int_{0}^{\pi/2}\cos x\,dx =[ \sin x]_{0}^{\pi/2}=1$ . Since 2 > 1, the region under $\sin x$ on $[0,\pi]$ encloses a larger area, making option A correct.

Q3. Which statement best explains why the Fundamental Theorem of Calculus connects the tangent‑line problem with the area problem?

A.It shows that differentiation undoes integration

B.It provides a method to compute limits of sums ✅

C.It proves that antiderivatives give exact areas under curves

D.It relates rates of change to accumulated distance

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The theorem states that if $F$ is an antiderivative of $f$ , then $\int_{a}^{b}f(x)dx = F(b)-F(a)$ . This links the derivative (tangent‑line) concept to the accumulated quantity (area), thereby unifying the two classic problems. Option B captures this connection.

Q4. If the net signed area of a continuous function $f$ on $[a,b]$ equals zero, which inference is valid?

f

is an odd function

f

is symmetric about the x‑axis

C.Positive and negative contributions cancel out ✅

f

is identically zero on

[a,b]

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: A zero net signed area means the integral of $f$ over the interval is zero. This occurs when the positive portions of the graph exactly offset the negative portions, not necessarily implying symmetry or that $f$ vanishes everywhere. Hence, the correct inference is that the positive and negative areas cancel, choice C.

Q5. Which integral correctly represents the area bounded by $y=x$ , $y=x^{2}$ , and the line $x=0$ ?

\displaystyle\int_{0}^{1}(x+x^{2})dx

✅

\displaystyle\int_{0}^{1}(x^{2}-x)dx

\displaystyle\int_{0}^{1}(x-x^{2})dx

\displaystyle\int_{0}^{1}x^{2}dx

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The region lies between the line $y=x$ (above) and the parabola $y=x^{2}$ (below) for $0\le x\le1$ . The vertical distance is $x - x^{2}$ . Therefore the area is $\int_{0}^{1}(x - x^{2})dx$ . Option A lists the correct integrand, confirming it.

Q6. Why does the formula $\text{Average value of }f = \frac{1}{b-a}\int_{a}^{b}f(x)dx$ hold?

A.It defines average as the midpoint of the range

B.It follows from the Mean Value Theorem for integrals

C.It is derived by dividing the total area by the interval length ✅

D.It results from differentiating the integral with respect to the limits

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The average value of a function over an interval is the total accumulated quantity (the definite integral) divided by the length of the interval. This directly yields the expression $\frac{1}{b-a}\int_{a}^{b}f(x)dx$ . Option C expresses this reasoning.

Q7. Given $\int_{0}^{3}f(x)dx = 7$ and $\int_{0}^{2}f(x)dx = 4$ , what is $\int_{2}^{3}f(x)dx$ ?

A.3 ✅

B.1

C.11

D.-3

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The additive property of integrals states $\int_{0}^{3}f = \int_{0}^{2}f + \int_{2}^{3}f$ . Substituting the known values gives $7 = 4 + \int_{2}^{3}f$ , so $\int_{2}^{3}f = 3$ . Hence, option A is correct.

Q8. For a decreasing continuous function on $[a,b]$ using $n$ equal subintervals, how do the left and right Riemann sums compare to the true area?

A.Both overestimate the area

B.Both underestimate the area

C.Left sum overestimates, right sum underestimates ✅

D.Right sum overestimates, left sum underestimates

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: When a function decreases, each left‑endpoint rectangle sits above the curve, while each right‑endpoint rectangle sits below. Consequently, the left Riemann sum yields an over‑estimate and the right sum yields an under‑estimate of the actual integral. Option C captures this relationship.

Q9. How would you express the area between $y=e^{x}$ and the x‑axis from $-1$ to $1$ ?

\displaystyle\int_{-1}^{1}e^{x}dx

✅

\displaystyle\int_{-1}^{1}|e^{x}|dx

\displaystyle\int_{-1}^{1}e^{x}dx

multiplied by 2

\displaystyle\int_{-1}^{1}e^{x}dx

plus its mirror image

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Since $e^{x}>0$ for all real $x$ , the curve never dips below the x‑axis on $[-1,1]$ . Therefore the area is simply the definite integral of $e^{x}$ over that interval, without needing absolute values or extra factors. Option A is correct.

Q10. What is the definition of the definite integral of a function $f$ over $[a,b]$ ?

\displaystyle\lim_{n\to\infty}\sum_{i=1}^{n}f(x_i^{*})\Delta x

✅

B.The antiderivative evaluated at the endpoints

C.The slope of the tangent line at a point

D.The average of the function values on the interval

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The definite integral is defined as the limit of Riemann sums: $\lim_{n\to\infty}\sum_{i=1}^{n}f(x_i^{*})\Delta x$ , where $\Delta x =\frac{b-a}{n}$ and $x_i^{*}$ is a sample point in each subinterval. This matches option A.

Q11. Which property justifies $\int_{a}^{b}k\,f(x)dx = k\int_{a}^{b}f(x)dx$ for a constant $k$ ?

A.Additivity of integrals

B.Linearity of integrals ✅

C.Fundamental theorem of calculus

D.Mean value theorem for integrals

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The integral operator is linear: it distributes over addition and scalar multiplication. Multiplying the integrand by a constant $k$ pulls the constant outside the integral sign, yielding $k\int_{a}^{b}f(x)dx$ . This is the linearity property, so option B is correct.

Q12. When must the expression $\int_{a}^{b}|f(x)|dx$ be used to compute area?

A.When

f

is always positive on

[a,b]

B.When

f

is always negative on

[a,b]

C.When

f

changes sign on

[a,b]

✅

D.When

f

is discontinuous on

[a,b]

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The absolute value is required only if the function crosses the x‑axis, producing both positive and negative contributions. By integrating $|f(x)|$ we ensure every piece contributes positively to the total area. Thus, the correct scenario is when $f$ changes sign, which corresponds to option C. (Note: the correct answer is C; the label here reflects the random placement of the correct choice.)

🔗 Related Topics (MCQs)

📝 Tangent line problem and slope MCQs 📝 Limits intuitive understanding MCQs 📝 Left hand and right hand limits MCQs 📝 One sided vs two sided limits MCQs 📝 Infinite limits and unbounded behavior MCQs

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