4.3
定積分
閉区間上に定義された実数値関数を考えます。微積分における基本的な目的の一つは、そのような関数のグラフとx軸で囲まれた面積を求めることです。正確な計算が容易にできない場合は、区間を有限個の等しい部分区間に分割することで、この面積を推定することができます。各部分区間は、その部分区間の長さを幅とする長方形…
閉区間の関数を考えます。曲線下の面積を近似するには、区間を等しい部分区間に分割し、等幅の長方形を形成します。
部分区間内の任意の点を選び、その関数値を長方形の高さとして使います。高さに幅を掛けると、その長方形の面積が得られます。曲線下の総面積はすべての長方形の面積を和にして近似し、リーマン和と呼ばれます。
与えられた区間上の関数の定積分は、部分区間の数が無限大に近づき、各長方形の幅が無限小になるときのリーマン和の極限として定義されます。ただし、この極限が存在する限ります。
例えば、速度が変化する車を考えます。移動時間を等しい時間間隔に割ります。
各区間で任意の時間点を選び、対応する速度を取る。速度に区間幅を掛けて移動距離を推定します。
これらの和はリーマン和を形成し、この例では曲線下の面積は物理的に車が走行した距離として解釈されます。速度が正のままであれば、速度関数をこれらの限界内で積分することで総距離を求めます。
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Q1: How does dividing an interval into subintervals help approximate area under a curve?
Dividing a closed interval into equal subintervals creates rectangles with uniform width. By selecting an arbitrary point within each subinterval and using the function value there as the rectangle's height, you can calculate each rectangle's area by multiplying height by width. Summing all rectangle areas produces a Riemann sum, which approximates the total area under the curve.
Q2: What is the relationship between Riemann sums and definite integrals?
A definite integral is defined as the limit of Riemann sums as the number of subintervals approaches infinity and each rectangle's width becomes infinitesimally small. This limiting process refines the approximation into an exact value representing the area under the curve over a specified interval, provided the limit exists.
Q3: How can you use integration to find the distance traveled by a moving object?
When a car travels with varying velocity, divide the total travel time into equal intervals and estimate distance in each interval by multiplying velocity at a chosen moment by the interval width. Summing these distances creates a Riemann sum. When velocity remains positive, integrating the velocity function over the time interval yields the exact total distance traveled.
Q4: Why does accuracy improve when you increase the number of subintervals?
As the number of subintervals increases, their widths decrease, making each rectangle's height more representative of the function's actual behavior over that interval. This finer partition reflects the curve's shape more closely, reducing the gap between the approximation and the true area, thereby improving accuracy.
Q5: What does it mean for a rectangle's height to be determined by an arbitrary point within a subinterval?
Within each subinterval, you can choose any point and evaluate the function at that location. The resulting function value becomes the rectangle's height for that subinterval. This flexibility in point selection is fundamental to constructing Riemann sums and demonstrates how different choices can yield different approximations of the area.
Q6: How does the definite integral represent a limit process?
The definite integral emerges when the partition of an interval becomes infinitely fine. As the number of rectangles grows without bound and each rectangle's width approaches zero, the Riemann sum converges to a limiting value. This limiting value represents the exact area under the curve and defines the definite integral.
Q7: What physical interpretation does the area under a velocity curve represent?
The area under a velocity-time curve represents the total distance traveled during that time interval. When velocity remains positive throughout, integrating the velocity function over the specified time limits yields the exact distance covered. This connection between integration and motion demonstrates how calculus models real-world phenomena.