69. Evaluate
∫₀¹ x dx + ∫₁² (2 − x) dx
Evaluate ∫₀¹ x dx + ∫₁² (2 − x) dx – Detailed Solution
The present problem consists of the sum of two definite integrals over different intervals. Although the question appears straightforward, it requires careful application of the Fundamental Theorem of Calculus and proper evaluation of the limits. Such questions are common in objective examinations because they assess whether students understand definite integration rather than simply memorizing integration formulas.
Interestingly, this problem can also be solved geometrically by interpreting each integral as the area under a straight line. Understanding both analytical and geometrical approaches strengthens conceptual clarity and helps solve similar questions much faster during competitive examinations.
Correct Answer
Answer = 1
Understanding the Concept of Definite Integration
A definite integral represents the accumulated value of a function over a specified interval. Geometrically, it represents the signed area between the graph of the function and the x-axis over the given interval. When the function remains above the x-axis, the definite integral equals the area enclosed by the curve and the x-axis.
In the present question, both functions are linear and remain non-negative over their respective intervals. Therefore, each integral represents a positive area.
Step 1: Evaluate the First Integral
The first integral is
∫₀¹ x dx
The integral of x is
x²/2.
Applying the limits,
= [x²/2]₀¹
= (1²/2) − (0²/2)
= 1/2 − 0
= 1/2
Thus, the value of the first definite integral is
1/2.
Step 2: Evaluate the Second Integral
The second integral is
∫₁² (2 − x) dx
Integrating each term separately,
∫(2 − x) dx
= 2x − x²/2
Now apply the limits.
= [2x − x²/2]₁²
Substituting the upper limit,
= 2(2) − (2²/2)
= 4 − 2
= 2
Substituting the lower limit,
= 2(1) − (1²/2)
= 2 − 1/2
= 3/2
Therefore,
= 2 − 3/2
= 1/2
Hence,
∫₁² (2 − x) dx = 1/2.
Step 3: Add Both Integrals
The required value is
= 1/2 + 1/2
= 1
Therefore, the value of the given expression is
1.
Mathematical Verification
First Integral:
∫₀¹ x dx = 1/2 ✔
Second Integral:
∫₁² (2 − x) dx = 1/2 ✔
Total:
1/2 + 1/2 = 1 ✔
The calculation is completely verified.
Geometrical Interpretation
This problem becomes even simpler when viewed geometrically.
The graph of y = x between x = 0 and x = 1 forms a right-angled triangle with
- Base = 1
- Height = 1
Therefore, its area is
½ × 1 × 1 = 1/2.
Similarly, the graph of y = 2 − x between x = 1 and x = 2 also forms another right-angled triangle having the same base and height.
Hence, its area is also
½ × 1 × 1 = 1/2.
The total area enclosed is therefore
1/2 + 1/2 = 1.
This geometrical interpretation provides an elegant shortcut and is often useful in multiple-choice examinations.
Alternative Method
Instead of performing integration, recognize that both integrals represent the areas of two congruent right triangles. Since each triangle has an area of 1/2, the answer can immediately be written as
1/2 + 1/2 = 1.
This approach saves valuable time during competitive examinations.
Related Practice Example
Evaluate
∫₀² x dx
Integrating,
= [x²/2]₀²
= (4/2) − 0
= 2.
Geometrically, this is the area of a triangle having base 2 and height 2.
Area = ½ × 2 × 2 = 2.
This example reinforces the relationship between definite integrals and geometrical area.
Key Takeaways
Whenever a definite integral involves simple linear functions, first determine whether direct integration or geometric interpretation provides the faster solution. For straight-line graphs, the area method is often significantly quicker than algebraic integration. Developing this insight can greatly improve speed and accuracy in competitive examinations.
Final Answer
The first integral equals 1/2, the second integral also equals 1/2.
Therefore,
∫₀¹ x dx + ∫₁² (2 − x) dx = 1.
Final Answer: 1
